![xii Preface
the algebra of all bounded linear operators on Hand K(H) for the ideal of L(H)
consisting of the compact operators. In particular, the property of being Fredholm
is invariant with respect to compact perturbations. Thus, no finite part of the
matrix representation (aij kjEZ of A E L(H) is responsible for the Fredholmness
of A, and the whole information on the Fredholm properties of A is hidden at
infinity, i.e., in the asymptotic behavior of the entries aij .
Limit operators. How can one draw information from infinity? A convenient way
is to fix a basis vector ek, to shift the operator A f---+ V-nAVn, and to observe the
evolution of the vectors V-nAVnek as n tends to ±oo. This has to be done for
each basis vector ek, which amounts to considering the behavior of the sequence
(V-nAVn) for large n with respect to the strong convergence of operators.
Assume for a moment that we are in the lucky case where the entries of each
diagonal of A stabilize at infinity (i.e., where the limits limi-doo aHk,i exist for
every integer k). Then the strong limits of the sequence (V-nAVn) as n -+ +00
and n -+ -00 exist, and these limits tell us exactly how the operator looks at
infinity. What happens in the general situation where the entries of each diagonal
are allowed to form an arbitrary bounded sequence? Then we cannot expect that
the sequence (V-nAVn) converges strongly but, hopefully, certain subsequences
will still do. Indeed, using a Cantor diagonal argument, we will even get that if A
is band-dominated, then every sequence h : N -+ Z which tends to (plus or minus)
infinity possesses a subsequence g : N -+ Z such that the sequence of the operators
V-g(n) AVg(n)
converges strongly as n -+ 00. The strong limit of this sequence is called the limit
operator of A with respect to the sequence g, and the set of all limit operators of a
given operator A is called the operator spectrum of A. The crucial and surprising
point is that the operator spectrum of a band-dominated operator contains exactly
the information from infinity which is needed to decide whether the operator is
Fredholm or not. The precise statement is: a band-dominated operator is Fredholm
if, and only if, its limit operators are invertible and if the norms of their inverses
are uniformly bounded.
Contents ofthe book. We will not restrict our attention to band-dominated oper-
ators on 12(Z); rather we consider band-dominated operators on IP(ZN, X) where
N is a positive integer, 1 ~ P ~ 00, and where X is a complex Banach space. The
main reason for this is that, after a suitable discretization, functions in LP(JRN)
become sequences in IP(ZN, X) with X = LP([O, l]N), and that a related dis-
cretization, applied to (wide classes of) convolution and pseudodifferential opera-
tors on LP(JRN), indeed produces band-dominated operators on the discrete space
IP(ZN, X). Thus, the theory of band-dominated operators which will be devel-
oped in the first two chapters can immediately be applied to convolution and
pseudodifferential operators to reproduce some known facts and to uncover some
new properties of these and other operators.](https://image.slidesharecdn.com/25584812-250516083312-c8643284/85/Limit-Operators-And-Their-Applications-In-Operator-Theory-Vladimir-Rabinovitch-17-320.jpg)
![1.1. Generalized compactness
such that
k + r(i) « k + r(il and k + r Ul « k + r(j+1)
1 1+1 4 1
for all 1 ::; I ::; 3, 1 ::; i ::; m and 1 ::; j ::; m - 1 and such that
7
IIPk+ri)AQk+r~i)11 < E/IIA-1
11
2
, (1.7)
IIQk+rii)APk+r~i)11 < E/IIA-1
11
2
, (1.8)
IIPk+rii)AQk+r~i) II < E/IIA-1
11
2
, (1.9)
IIQk+ri+l)APk+r~i)11 < E/IIA-1
11
2
. (1.10)
That is, given riil we choose r~il > riil such that k +r~i) » k +riil and that (1.7)
holds, then r~i) > r~i) such that k +r~il » k +r~i) and that (1.8) is satisfied, then
r~i) > r~i) which fulfills k + r~il » k + r~il and (1.9), and finally riHll > r~il such
that k + riHll » k + r~il and that (1.10) is valid.
Let n» k + r~ml. We set
U· '= (k + r(il k + r(ilj V; '= (k + r(il k + r(ilj
t· l ' 3' t· 2' 4'
TT' ' - [0 k (il] U' '- [0 k (il]
vi'- , + r2 , i ' - ' + r3 .
Then, since n » k + r~ml » k + r~il for all i,
Pk Pv/A- 1
Qn
PkA-1Pu~APv,A-1Qn + PkA-1Qu,APv,A-1Qn (1.11)
t '( t t
with
IlPkA-1Qu:APv/A-1Qnll < C(C + 1) IIA-
1
11
2
1IQu:APv/ II
C(C + 1) IIA-11121IQk+rii)APk+r~i) II
< C(C + I)E (1.12)
due to (1.8) (observe that IIQul1 = III - Pull::; 1+C). Further, since n» k+r~il,
PkA-1Pu:APv/A- 1Qn
- PkA-1 Pu,AQv,A-1Qn
t t
- PkA-1Pui AQv/A-1Qn - PkA-1PU:Ui AQV/A-1Qn
= - PkA- I PUiAPviA-IQn - PkA-1PUiAQk+r~i)A-1Qn
- PkA- I Pu'uAPvA-IQn - PkA- I PU'UAQk+ (i)A-IQn
t t t i t T
4
= - PkA- I PUiAPViA-1Qn - PkA-I PU:AQk+r~i)A-IQn (1.13)
- PkA- I PU:UiAPviA-IQn'](https://image.slidesharecdn.com/25584812-250516083312-c8643284/85/Limit-Operators-And-Their-Applications-In-Operator-Theory-Vladimir-Rabinovitch-27-320.jpg)
![1.4. Comments and references
1.4 Comments and references
29
The first appearance of limit operators is in Favard's paper [53], where they are
used to verify the existence of almost-periodic solutions of ordinary differential
equations with almost-periodic coefficients. Later, Muhamadiev [109, 110] applied
limit operators to the question of solvability of elliptic partial differential equations
in IRn
. The limit operators method has been developed further in the papers [93,
95, 94, 118, 126, 137] for the study of the Fredholm property of wide classes of
pseudodifferential operators and convolution operators on IRn and zn. Note also the
paper [27], where the limit operators method has been applied to the computation
of the essential spectrum of singular integral operators on Carleson curves acting
in general weighted L2
-spaces. See also the monograph [72] for lots of applications
in numerical analysis. We will give impressions of these results in Chapters 3, 4
and 6. Observe that in all of these papers, the method of limit operators is applied
to a concrete class of operators acting on a concrete Banach space.
Generalized notions of compactness have been introduced in [31, 52, 149],
for example. Theorem 1.1.9 and its proof are (with minor modifications) taken
from Kozak and Simonenko [87]. The original formulation of their result says that
the inverse of an operator of local type is of local type again. It has been already
employed in [149] to study the stability of approximation methods on spaces with
supremum norm.
There are other algebras besides L(E, P) which can be associated with an ap-
proximate projection P in a natural way. A good candidate would be OLT(E, P),
the algebra of all operators A E L(E) with IlPnA - APnII ---> 0 as n ---> 00. One
easily checks that OLT(E, P) is a closed subalgebra of L(E) and that K(E, P) ~
OLT(E, P) ~ L(E, P). Moreover, it is evident that the algebra OLT(E, P) is in-
verse closed in L(E). So, working with operators in OLT(E, P) seems to be much
easier than working with operators from L(E, P). We give L(E, P) preference
over OLT(E, P) for at least two reasons: L(E, P) is not only the largest algebra
for which the P-compact operators form an ideal; this algebra is also (in contrast
to OLT(E, P)) invariant with respect to the substitution of P by an equivalent
approximate projection. Thus, when dealing with operators in L(E, P), we can
switch over between equivalent approximate projections and choose that one which
fits to our actual purposes.](https://image.slidesharecdn.com/25584812-250516083312-c8643284/85/Limit-Operators-And-Their-Applications-In-Operator-Theory-Vladimir-Rabinovitch-49-320.jpg)
![34 Chapter 2. Band-dominated Operators
Proposition 2.1.2 For a E lOO(ZN, L(X)), the operator aI belongs to L(EOO , P).
This operator belongs to K(EOO , P) if and only if a E cO(ZN, L(X)).
Proof. Let K E K(EOO, P). Since aI commutes with every operator Qn, we have
IlaKQnll:::; IlaIllllKQnl1 ~ 0 and IIKaQnl1 = IIKQnaIl1 :::; IIKQnllllaIl1 ~ 0
as n ~ 00. Similarly, IIQnKaIl1 ~ 0 and IIQnaKl1 ~ 0, whence aI E L(EOO, P).
For the second assertion observe that aI E K(EOO , P) if and only if, given
c > 0, there is an no E N such that IlaQnl1 :::; E: for all n 2: no or, equivalently,
such that Ila(x)11 :::; E: for all x with Ixl > no. The latter is equivalent to a E
cO(ZN, L(X)). 0
We proceed with two equivalent characterizations of multiplication operators.
Proposition 2.1.3 An operator A E L(EOO) is an operator of multiplication by a
function in lOO(ZN, L(X)) if and only if
ASm = SmA for allm E ZN. (2.2)
Proof. It is evident that (2.2) holds if A is a multiplication operator. For the reverse
implication, let A be subject to (2.2), and let En and Rn be defined as in Section
2.1.2. By MA we denote the operator of multiplication by the function
a: ZN ~ L(X), m ~ RmSmASmEm.
Evidently, a E lOO(ZN,L(X)), and for all m E ZN and all U = (Um)mEZN E Eoo,
we have
SmMAU = Ema(m)um = EmRmSmASmEmum = SmASmu = SmAu
due to (2.2). This necessarily implies that MAu = Au for all U E Eoo, i.e., A is a
multiplication operator. 0
Thus, an operator is an operator of multiplication if and only if its matrix repre-
sentation is a diagonal matrix. For another criterion, let t = (h, ... , tN) E ]RN,
consider the function et which is defined at x = (Xl, ... , XN) E ]RN by
(2.3)
and denote the restriction of a function a on ]RN onto ZN bya.
Theorem 2.1.4
(a) An operator A E L(E) is the operator of multiplication by a function in
lOO(ZN, L(X)) if and only if
(2.4)
(b) An operator A E L(E oo , P) is the operator of multiplication by a function in
lOO(ZN, L(X)) if and only if (2.4) holds.](https://image.slidesharecdn.com/25584812-250516083312-c8643284/85/Limit-Operators-And-Their-Applications-In-Operator-Theory-Vladimir-Rabinovitch-53-320.jpg)
![2.1. Band-dominated operators 35
Proof. Clearly, every multiplication operator satisfies (2.4). For the reverse direc-
tion, we claim that if an operator A E L(Eoo ) is subject to condition (2.4), then
AbPn = bAPn for every bounded continuous function b on IR.N
. (2.5)
Once this is verified, and if A E L(Eoo, P), we take the P-strong limit of both
sides of (2.5) which yields AbI = bA for all b and, hence, ASm = SmA for all m.
Then assertion (b) follows from Proposition 2.1.3. In case A E L(E), we can take
the usual strong limit in place of the P-strong one which yields the conclusion
without further restriction on A.
To verify (2.5), let £ denote the closure with respect to the supremum norm
of the linear hull of all functions et, and let M n := [-n, n]N. Then, due to (2.4),
aA = AM for all a E £. (2.6)
The WeierstraB' approximation theorem implies that the restriction of £ onto M n
is all of C(Mn ). Hence, if b is a bounded continuous function on IR.N then, for every
fixed n, there is a function a E £ such that alMn = bl Mn . Consequently,
AbPnf = Aapnf = aAPnf for all f E E oo (2.7)
due to (2.6). Further, the function a can be chosen such that it takes at x E ZNMn
an arbitrarily prescribed value. Since the left-hand side of (2.7) is independent of
a, this shows that the function APnf vanishes outside ZN n Mn. Hence,
AbPnf = aAPnf = bAPnf for all f E E oo
which proves our claim (2.5). 0
There are operators on L(loo(Z)) which satisfy (2.4), but which fail to be multi-
plication operators (see [91], Remark 2.1.9).
2.1.4 Band and band-dominated operators
The basic objects of our interest are band and band-dominated operators. They are
constituted by operators of multiplication and by the operators of shift introduced
in (2.1).
Definition 2.1.5 A band operator is a finite sum of the form La aaVa where a E
ZN and aa E loo(ZN,L(X)). A band-dominated operator is the uniform limit of
a sequence of band operators.
To justify this notation note that, in case X = C. and N = 1 and with respect to
the standard basis of E oo , band operators are given by matrices with finite band
width. Observe also that the class of band operators is independent of the concrete
space Eoo, whereas the class of band-dominated operators depends heavily on E oo .
We denote this class by AEoo.](https://image.slidesharecdn.com/25584812-250516083312-c8643284/85/Limit-Operators-And-Their-Applications-In-Operator-Theory-Vladimir-Rabinovitch-54-320.jpg)
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Title: Rakkautta ja politiikkaa: Huvinäytelmä 1:ssä näytöksessä
Author: Armas E. Turunen
Release date: September 17, 2016 [eBook #53068]
Language: Finnish
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*** START OF THE PROJECT GUTENBERG EBOOK RAKKAUTTA JA
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Limit Operators And Their Applications In Operator Theory Vladimir Rabinovitch
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- 7. Operator Theory: Advances and Applications Vol. 150 Editor: I. Gohberg Editorial Office: School of Mathematical Sciences Tel Aviv University Ramat Aviv, Israel Editorial Board: D. Alpay (Beer-Sheva) J. Arazy (Haifa) A. Atzmon (Tel Aviv) J. A. Ball (Blacksburg) A. Ben-Artzi (Tel Aviv) H. Bercovici (Bloomington) A. Bottcher (Chemnitz) K. Clancey (Athens, USA) L. A. Coburn (Buffalo) K. R. Davidson (Waterloo, Ontario) R. G. Douglas (College Station) A. Dijksma (Groningen) H. Dym (Rehovot) P. A. Fuhrmann (Beer Sheva) S. Goldberg (College Park) B. Gramsch (Mainz) G. Heinig (Chemnitz) J. A. Helton (La Jolla) M. A. Kaashoek (Amsterdam) H. G. Kaper (Argonne) S.1. Kuroda (Tokyo) P. Lancaster (Calgary) L. E. Lerer (Haifa) B. Mityagin (Columbus) V. V. Peller (Manhattan, Kansas) L. Rodman (Williamsburg) J. Rovnyak (Charlottesville) D. E. Sarason (Berkeley) I. M. Spitkovsky (Williamsburg) S. Treil (Providence) H. Upmeier (Marburg) S. M. Verduyn Lunel (Leiden) D. Voiculescu (Berkeley) H. Widom (Santa Cruz) D. Xia (Nashville) D. Yafaev (Rennes) Honorary and Advisory Editorial Board: C. Foias (Bloomington) P. R. Halmos (Santa Clara) 1. Kailath (Stanford) P. D. Lax (New York) M. S. Livsic (Beer Sheva)
- 8. Limit Operators and Their Applications in Operator Theory Vladimir Rabinovich Steffen Roch Bernd Silbermann Springer Basel AG
- 9. Authors: Vladimir Rabinovich Instituto Politecnico Nacional ESIME Zacatenco Avenida IPN Mexico, D,F. 07738 Mexico vladimicrabinovich@hotmail.com Bernd Silbermann Department of Mathematics Technical University of Chemnitz 09107 Chemnitz Oermany silbermn@mathematik.tu-chemnitz.de Steffen Roch Department of Mathematics Technical University of Darmstadt Schlossgartenstrasse 7 64289 Darmstadt Oermany roch@mathematik.tu-darmstadt.de 2000 Mathematics Subject Classification 47L80; 35S05, 47A53, 47030, 65R20 ISBN 978-3-0348-9619-1 ISBN 978-3-0348-7911-8 (eBook) DOI 10.1007/978-3-0348-7911-8 A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>. This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © Springer Basel AG 2004 Originally published by Birkhăuser Verlag AG 2004 Softcover reprint of the hardcover 1st edition 2004 Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF 00 Cover design: Heinz Hiltbrunner, Basel ISBN 3-7643-7081-5 987654321 www.birkhăuser-science.com
- 10. Contents Preface . 1 Limit Operators 1.1 Generalized compactness, generalized convergence 1.1.1 Compactness, strong convergence, Fredholmness 1.1.2 P-compactness .... 1.1.3 P-Fredholmness 1.1.4 P-strong convergence 1.1.5 Invertibility of P-strong limits 1.2 Limit operators . . . . . . . 1.2.1 Limit operators and the operator spectrum 1.2.2 Operators with rich operator spectrum. 1.3 Algebraization . 1.3.1 Algebraization by restriction 1.3.2 Symbol calculus . 1.4 Comments and references . . . . . . 2 Fredholmness of Band-dominated Operators 2.1 Band-dominated operators . 2.1.1 Function spaces on TLN . 2.1.2 Matrix representation . 2.1.3 Operators of multiplication 2.1.4 Band and band-dominated operators 2.1.5 Limit operators of band-dominated operators 2.1.6 Rich band-dominated operators . 2.2 P-Fredholmness of rich band-dominated operators . 2.2.1 The main theorem on P-Fredholmness .... 2.2.2 Weakly sufficient families of homomorphisms 2.2.3 Symbol calculus for rich band-dominated operators. 2.2.4 Appendix A: Second version of a symbol calculus 2.2.5 Appendix B: Commutative Banach algebras ..... Xl 1 1 4 10 11 15 17 17 19 23 24 25 29 31 31 32 33 35 40 43 45 45 51 53 56 59
- 11. vi Contents 2.3 Local P-Fredholmness: elementary theory . . . . . . 61 2.3.1 Local operator spectra and local invertibility 61 2.3.2 PR-compactness, PR-Fredholmness . . . . . 62 2.3.3 Local P-Fredholmness of band-dominated operators 64 2.3.4 Allan's local principle . . . . . . . . . . . . . . . . . 65 2.3.5 Local P-Fredholmness of band-dominated operators in the sense of the local principle . . . 69 2.3.6 Operators with continuous coefficients 72 2.4 Local P-Fredholmness: advanced theory . . 74 2.4.1 Slowly oscillating functions . . . . . 74 2.4.2 The maximal ideal space of SO(ZN) 79 2.4.3 Preliminaries on nets . . . . . . . . . 82 2.4.4 Limit operators with respect to nets 87 2.4.5 Local invertibility at points in Moo(SO(ZN)) 89 2.4.6 Fredholmness of band-dominated operators with slowly oscillating coefficients. . . . . . . 93 2.4.7 Nets vs. sequences . . . . . . . . . . . . . . . 94 2.4.8 Appendix A: A second proof of Theorem 2.4.27 . 95 2.4.9 Appendix B: A third proof of Theorem 2.4.27 . 100 2.5 Operators in the discrete Wiener algebra. . . . . . . . . 103 2.5.1 The Wiener algebra . . . . . . . . . . . . . . . . 103 2.5.2 Fredholmness of operators in the Wiener algebra 107 2.6 Band-dominated operators with special coefficients 111 2.6.1 Band-dominated operators with almost periodic coefficients 111 2.6.2 Operators on half-spaces. . . . . . . . . 113 2.6.3 Operators on polyhedral convex cones 119 2.6.4 Composed band-dominated operators on Z2 124 2.6.5 Difference operators of second order . . 128 2.6.6 Discrete Schrodinger operators . . . . . 131 2.7 Indices of Fredholm band-dominated operators 135 2.7.1 Main results. . . . . . . . . . . . . . . . 136 2.7.2 The algebra A(Z) as a crossed product. 138 2.7.3 The K1-group of A(Z) . 139 2.7.4 The K1-group of A± . . . . . . . . 142 2.7.5 Proof of Theorem 2.7.1 144 2.7.6 Unitary band-dominated operators 147 2.8 Comments and references . . . . . . . . . 150 3 Convolution Type Operators on lRN 3.1 Band-dominated operators on LP(lRN ) . 3.1.1 Approximate identities and P-Fredholmness . 3.1.2 Shifts and limit operators . 153 153 155
- 12. Contents 3.1.3 Discretization . 3.1.4 Band-dominated operators on LP(JRN) 3.2 Operators of convolution . 3.2.1 Compactness of semi-commutators . 3.2.2 Compactness of commutators .... 3.3 Fredholmness of convolution type operators 3.3.1 Operators of convolution type . 3.3.2 Fredholmness..... . 3.4 Compressions of convolution type operators 3.4.1 Compressions of operators in A(BUC(JRN ), Cp ) 3.4.2 Compressions to a half-space .... 3.4.3 Compressions to curved half-spaces. 3.4.4 Compressions to curved layers 3.4.5 Compressions to curved cylinders .. 3.4.6 Compressions to cones with smooth cross section 3.4.7 Compressions to cones with edges .... 3.4.8 Compressions to epigraphs of functions 3.5 A Wiener algebra of convolution-type operators . 3.5.1 Fredholmness of operators in the Wiener algebra 3.5.2 The essential spectrum of Schrodinger operators 3.6 Comments and references . vii 155 157 159 159 164 169 169 172 179 180 181 182 184 184 185 190 193 194 194 195 199 4 Pseudodifferential Operators 4.1 Generalities and notation . . . . . . . . . . . . 201 4.1.1 Function spaces and Fourier transform. 201 4.1.2 Oscillatory integrals . . . . . 203 4.1.3 Pseudodifferentialoperators....... 204 4.1.4 Formal symbols . . . . . . . . . . . . . . 205 4.1.5 Pseudodifferential operators with double symbols 206 4.1.6 Boundedness on L2(JRN) . . . . . . . . . . . . . . 207 4.1.7 Consequences of the Calderon-Vaillancourt theorem 210 4.2 Bi-discretization of operators on L2(JRN) . . 211 4.2.1 Bi-discretization . . . . . . . . . . . 211 4.2.2 Bi-discretization and Fredholmness . 213 4.2.3 Bi-discretization and limit operators 215 4.3 Fredholmness of pseudodifferential operators. 218 4.3.1 A Wiener algebra on L2 (JRN) . . . . . 218 4.3.2 Fredholmness of operators in W$(L2(JRN)) 222 4.3.3 Fredholm properties of pseudodifferential operators in 0 P88,0 . . . . . . . . . . . . . . . . . . . . . . 224
- 13. viii Contents 4.4 Applications. 228 4.4.1 Operators with slowly oscillating symbols 228 4.4.2 Operators with almost periodic symbols 230 4.4.3 Operators with semi-almost periodic symbols 233 4.4.4 Pseudodifferential operators of nonzero order 234 4.4.5 Differential operators. 236 4.4.6 Schrodinger operators 239 4.4.7 Partial differential-difference operators 242 4.5 Mellin pseudodifferential operators 243 4.5.1 Pseudodifferential operators with analytic symbols 243 4.5.2 Mellin pseudodifferential operators 247 4.5.3 Mellin pseudodifferential operators with analytic symbols 250 4.5.4 Local invertibility of Mellin pseudodifferential operators 251 4.6 Singular integrals over Carleson curves with Muckenhoupt weights 254 4.6.1 Carleson curves and Muckenhoupt weights . 254 4.6.2 Logarithmic spirals and power weights . 255 4.6.3 Curves and weights with slowly oscillating data . 257 4.6.4 Local representatives and local spectra of singular integral operators . 258 4.6.5 Singular integral operators on composed curves 262 4.7 Comments and references 265 5 Pseudodifference Operators 5.1 Pseudodifference operators. . . . . . . . . . . . . . 5.2 Fredholmness of pseudodifference operators .... 5.3 Fredholm properties of pseudodifference operators on weighted spaces . . . . . . . . . . . . 5.3.1 Boundedness on weighted spaces 5.3.2 Fredholmness on weighted spaces 5.3.3 The Phragmen-Lindelof principle 5.4 Slowly oscillating pseudodifference operators. 5.4.1 Fredholmness on lP-spaces ..... 5.4.2 Fredholmness on weighted spaces, Phragmen-LindelOf principle 5.4.3 Fredholm index for operators in OPSO 5.5 Almost periodic pseudodifference operators 5.6 Periodic pseudodifference operators . 5.6.1 The one-dimensional case . 5.6.2 The multi-dimensional case . 267 273 276 276 278 279 280 280 284 287 288 289 290 292
- 14. Contents 5.7 Semi-periodic pseudodifference operators .. 5.7.1 Fredholmness on unweighted spaces 5.7.2 Fredholmness on weighted spaces 5.7.3 Fredholm index . 5.8 Discrete Schrodinger operators . 5.8.1 Slowly oscillating potentials . 5.8.2 Exponential decay of eigenfunctions 5.8.3 Semi-periodic Schrodinger operators 5.9 Comments and references . 6 Finite Sections of Band-dominated Operators 6.1 Stability of the finite section method 6.1.1 Approximation sequences . 6.1.2 Stability vs. invertibility ... 6.1.3 Stability vs. Fredholmness .. 6.2 Finite sections of band-dominated operators on Zl and Z2 6.2.1 Band-dominated operators on Zl: the general case 6.2.2 Band-dominated operators on Zl: slowly oscillating coefficients . . . . . . . . . . . . . 6.2.3 Band-dominated operators on Z2 . . . .... 6.2.4 Finite sections of convolution type operators 6.3 Spectral approximation . 6.3.1 Weakly sufficient families and spectra . 6.3.2 Interlude: Spectra of band-dominated operators on Hilbert spaces . . . . . . . . 6.3.3 Asymptotic behavior of norms 6.3.4 Asymptotic behavior of spectra 6.4 Fractality of approximation methods . 6.4.1 Fractal approximation sequences 6.4.2 Fractality and norms . 6.4.3 Fractality and spectra . 6.4.4 Fractality of the finite section method for a class of band-dominated operators 6.5 Comments and references . 7 Axiomatization of the Limit Operators Approach 7.1 An axiomatic approach to the limit operators method 7.2 Operators on homogeneous groups 7.2.1 Homogeneous groups .. 7.2.2 Multiplication operators 7.2.3 Partition of unity ... 7.2.4 Convolution operators 7.2.5 Shift operators .... ix 293 293 296 297 297 298 299 301 302 304 304 306 307 312 313 315 318 320 321 322 326 327 328 332 333 335 336 339 342 345 361 361 362 363 364 365
- 15. x Contents 7.3 Fredholm criteria for convolution type operators with shift. 368 7.3.1 Operators on homogeneous groups 368 7.3.2 Operators on discrete subgroups 372 7.4 Comments and references . . . . . . . . 373 Bibliography Index .... 375 387
- 16. Preface This text has two goals. It describes a topic: band and band-dominated operators and their Fredholm theory, and it introduces a method to study this topic: limit operators. Band-dominated operators. Let H = [2(Z) be the Hilbert space of all squared summable functions x : Z -+ <C, i f---> Xi provided with the norm IIxl1 2 := L IXiI2 . iEZ It is often convenient to think of the elements x of [2(Z) as two-sided infinite sequences (Xi)iEZ. The standard basis of [2(Z) is the family of sequences (ei)iEZ where ei = (... ,0,0, 1,0,0, ...) with the 1 standing at the ith place. Every bounded linear operator A on H can be described by a two-sided infinite matrix (aij )i,jEZ with respect to this basis, where aij = (Aej, ei)' The band operators on H are just the operators with a matrix representation of finite band-width, i.e., the operators for which aij = 0 whenever Ii - jl > k for some k. Operators which are in the norm closure of the algebra of all band operators are called band-dominated. Needless to say that band and band- dominated operators appear in numerous branches of mathematics. Archetypal examples come from discretizations of partial differential operators. It is easy to check that every band operator can be uniquely written as a finite sum L dkVk where the dk are multiplication operators (i.e., they are given by a diagonal matrix with respect to the standard basis), and where the Vk are the shift operators ej f---> ej+k. Conversely, every finite sum of this form is a band operator. This equivalence allows us to think of band operators as being composed of two kinds of generators - multiplication operators and shift operators. Fredholmness. We will be mainly concerned with the Fredholm properties of band-dominated operators. A bounded linear operator A on H is called a Fredholm operator if both its kernel {x E H : Ax = O} and its cokernel H/(AH) are finite- dimensional linear spaces. Equivalently, an operator A is Fredholm if its coset A + K(H) is invertible in the Calkin algebra L(H)/K(H) where L(H) stands for
- 17. xii Preface the algebra of all bounded linear operators on Hand K(H) for the ideal of L(H) consisting of the compact operators. In particular, the property of being Fredholm is invariant with respect to compact perturbations. Thus, no finite part of the matrix representation (aij kjEZ of A E L(H) is responsible for the Fredholmness of A, and the whole information on the Fredholm properties of A is hidden at infinity, i.e., in the asymptotic behavior of the entries aij . Limit operators. How can one draw information from infinity? A convenient way is to fix a basis vector ek, to shift the operator A f---+ V-nAVn, and to observe the evolution of the vectors V-nAVnek as n tends to ±oo. This has to be done for each basis vector ek, which amounts to considering the behavior of the sequence (V-nAVn) for large n with respect to the strong convergence of operators. Assume for a moment that we are in the lucky case where the entries of each diagonal of A stabilize at infinity (i.e., where the limits limi-doo aHk,i exist for every integer k). Then the strong limits of the sequence (V-nAVn) as n -+ +00 and n -+ -00 exist, and these limits tell us exactly how the operator looks at infinity. What happens in the general situation where the entries of each diagonal are allowed to form an arbitrary bounded sequence? Then we cannot expect that the sequence (V-nAVn) converges strongly but, hopefully, certain subsequences will still do. Indeed, using a Cantor diagonal argument, we will even get that if A is band-dominated, then every sequence h : N -+ Z which tends to (plus or minus) infinity possesses a subsequence g : N -+ Z such that the sequence of the operators V-g(n) AVg(n) converges strongly as n -+ 00. The strong limit of this sequence is called the limit operator of A with respect to the sequence g, and the set of all limit operators of a given operator A is called the operator spectrum of A. The crucial and surprising point is that the operator spectrum of a band-dominated operator contains exactly the information from infinity which is needed to decide whether the operator is Fredholm or not. The precise statement is: a band-dominated operator is Fredholm if, and only if, its limit operators are invertible and if the norms of their inverses are uniformly bounded. Contents ofthe book. We will not restrict our attention to band-dominated oper- ators on 12(Z); rather we consider band-dominated operators on IP(ZN, X) where N is a positive integer, 1 ~ P ~ 00, and where X is a complex Banach space. The main reason for this is that, after a suitable discretization, functions in LP(JRN) become sequences in IP(ZN, X) with X = LP([O, l]N), and that a related dis- cretization, applied to (wide classes of) convolution and pseudodifferential opera- tors on LP(JRN), indeed produces band-dominated operators on the discrete space IP(ZN, X). Thus, the theory of band-dominated operators which will be devel- oped in the first two chapters can immediately be applied to convolution and pseudodifferential operators to reproduce some known facts and to uncover some new properties of these and other operators.
- 18. Preface xiii The inclusion of the 'exotic' case p = 00 and the consideration of sequences with values in infinite-dimensional Banach spaces involve some subtleties. For ex- ample, we are limited when working with strong convergence of operators, because the projections Pn : [P(Z) ~ [P(Z) which replace all entries of a sequence (Xk) with Ikl > n by zero converge strongly to the identity operator if and only if p i:- 00. Another point is that the notions of Fredholmness and of invertibility at infinity (which are synonym for operators on spaces of sequences which take values in a finite-dimensional Banach space) become basically different for operators on spaces of sequences with values in infinite-dimensional Banach spaces. The applications we have in mind suggest to give preference to the aspect of invertibility at infinity over Fredholmness. So we start in the first chapter with modifying the standard concepts of strong convergence, compactness and Fredholmness by introducing the notions of P-strong convergence, P-compactness and P-Fredholmness. Here, P is a given approximate identity, for instance, the sequence (Pn ) we encountered in the pre- ceding paragraph. Based on these P-notions, we introduce the general concept of a limit operator in Section 1.2. In case 1 < p < 00 and dim X < 00, all P-notions reduce to their usual mean- ings. Thus, readers who are exclusively interested in band-dominated operators on scalar-valued sequences can skip the first part of this chapter. It is perhaps also a good advice for a first reading of Chapters 1 and 2 to skip Section 1.1, to ignore the P, and to set X = C in what follows. The second chapter is the heart of the book. Here we introduce band-domi- nated operators on the spaces [P(71P, X) and prove that they possess sufficiently many limit operators in the sense that a band-dominated operator is P-Fredholm if and only if all of its limit operators are invertible and if the norms of their inverses are uniformly bounded. The status of the uniform invertibility hypothesis is not really evident at this moment. On the one hand, it might be quite hard to check the uniform boundedness of the norms of the inverses of a family of operators explicitly; so the condition is very unpleasant from this practical point of view. On the other hand, we do not know any example of a band-dominated operator, all limit operators of which are invertible, but which fails to be P-Fredholm. Moreover, it turns out that there are large and important classes of band-dominated operators for which we can prove that the uniformity of the invertibility is indeed redundant. In particular, this happens for band-dominated operators in the Wiener algebra (which includes all band operators) as well as for band-dominated operators with slowly oscillating coefficients. These results will also be presented in Chapter 2. Partially, these results are based on the compatibility of the limit operators method with another local principle, which is due to Allan and Douglas. Thus, a large part of the second chapter is devoted to the study of the relations between these local theories. The last part of Chapter 2 deals with the problem of calculating the index of a Fredholm band-dominated operator A in terms of its limit operators. Here we restrict ourselves to band-dominated operators on [2(Z) with scalar-valued coeffi-
- 19. XIV Preface cients. Under these assumptions, we get an index formula of astonishing simplicity. For, choose an arbitrary limit operator B+ of A with respect to a sequence tending to +00 as well as an arbitrary limit operator B_ of A with respect to a sequence tending to -00. Then the index of A is the sum of the local indices of B± at ±oo, In Chapters 3 and 4, we are going to specify the results on Fredholmness obtained for general band-dominated operators to convolution operators and pseu- dodifferential operators on LP(lRN ), respectively. The key steps are to embed these operators into suitable operator algebras of Wiener type and to discretize in an appropriate manner these operators to get band-dominated operators on a discrete lP-space of vector-valued sequences. A similar approach is chosen in Chapter 5 in order to illustrate the applicability of the limit operators method to study the Fredholmness of pseudodifference operators on lP(ZN)-spaces with weight. Par- ticular attention is paid to Phragmen-Lindelof type theorems on the exponential decay at infinity of solutions to pseudodifference equations, to the description of the essential spectrum of discrete Schrodinger operators, and to the decay of their eigenfunctions at infinity. Chapter 6 shifts the attention from analysis to numerical analysis. We con- sider the finite section method for the approximate solution of equations with band-dominated system matrices. The basic observation here is that the sequence of the finite sections of a band-dominated operator on ZN can be interpreted as a band-dominated operator on ZN+l. Moreover, the sequence is stable if and only if the corresponding operator is P-Fredholm. Thus, the results from Chapter 2 apply immediately to yield stability results for the finite section method. In the Hilbert space case p = 2, these stability results will be further used to derive characteri- zations of the asymptotic behavior of the norms, condition numbers, eigenvalues, pseudo-eigenvalues, and Rayleigh quotients of the finite section matrices. So far we have only discussed the application of the limit operators method to the Fredholm theory of band-dominated (and related) operators. The goal of the final Chapter 7 is to indicate that the range of the applicability of the limit operators method is much larger. So we will develop an axiomatic scheme which covers most applications of the limit operators method. As concrete examples we consider the Fredholmness of convolution operators as well as of convolutions combined with (non-Carleman) shifts on the Heisenberg group. Preliminaries. We assume that the reader has basic knowledge in linear Func- tional Analysis. For Chapters 4 and 7, a first acquaintance with pseudodifferential operators and with non-commutative harmonic analysis would be helpful. Acknowledgements. The authors are grateful to their friends and colleagues Al- brecht Bottcher, Marko Lindner and Alexander Rogozhin who read large parts of the manuscript, suggested many improvements and also pointed out a few mistakes
- 20. Preface xv to us. We also thank the Publishers for their kind and stimulating co-operation. Finally, we would like to appreciate our sincere thanks to the German Research Foundation (DFG) and to the CONACYT which supported the research on this subject as well as the writing of this book by several travel and research grants (DFG grants 436 RUS 17/67/98 and 436 RUS 17/24/01, and CONACYT project 32424-E). This book would not exist without the generous support by these insti- tutions.
- 21. Chapter 1 Limit Operators 1.1 Generalized compactness, generalized convergence We start with recalling how the class of the compact operators on a Banach space determines both the strong operator topology and the set of the Fredholm opera- tors. This characterization is then taken as a starting point to introduce generalized notions of strong convergence and Fredholmness. 1.1.1 Compactness, strong convergence, Fredholmness Let E be a Banach space with norm II.IIE and with Banach dual E*. All Banach spaces occurring in this book are supposed to be linear spaces over the field <C of the complex numbers. By L(E) we denote the Banach algebra of the bounded linear operators on E with norm IIAIIL(E):= sup IIAxIIE/llxIIE. xEE{O} We further let K(E) stand for the closed ideal of the compact operators in L(E). The identity operator on E will be denoted by I. Finally, we will write Ker A and ImA for the kernel {x E E : Ax = O} and the range {Ax: x E E} of A, respectively. Strong convergence. A sequence (An) of operators An E L(E) is said to converge strongly if the sequence (Anx) converges in the norm of E for each x E E. Then Ax := lim Anx defines an operator A on E which is called the strong limit of the sequence (An) and which we denote by s-limAn. One also says that the An converge strongly to A and writes An ---+ A strongly. In general, the adjoint sequence (A~) of a strongly convergent sequence (An) fails to be strongly convergent. Thus, the sequence (An) is said to converge *- strongly if both sequences (An) and (A~) converge strongly on E and E*, respec- tively. In this case, s-lim A~ = (s-lim An )*.
- 22. 2 Chapter 1. Limit Operators Theorem 1.1.1 (Banach-Steinhaus) Let the sequence (An) of bounded linear opera- tors on E be strongly convergent to an operator A. Then A is a bounded and linear operator on E, the sequence (An) is uniformly bounded, and IIAIIL(E) :S liminf IIAnIIL(E)' We agree upon calling a non-empty set M C L(E) of operators uniformly invertible if the operators in M are invertible and if the norms of their inverses are uniformly bounded. Proposition 1.1.2 (a) If An ----; A and B n ----; B strongly, then An + Bn ----; A + Band AnBn ----; AB strongly. (b) If An ----; A strongly, and if the operators An are uniformly invertible, then A has a trivial kernel and a closed range, and A;;-l A ----; I strongly. (c) If An ----; A *-strongly, and if the operators An are uniformly invertible, then A is invertible, and A;;: 1 ----; A-I * -strongly. Proof. The proof of (a) is straightforward. It makes use of the uniform boundedness of the sequence (An) due to the Banach-Steinhaus theorem. The last assertion of (b) is a consequence of the estimate and of the strong convergence of An to A. Thus, letting n go to infinity in IIA;;-lAxil :S CIIAxll, we obtain Ilxll :S CllAxl1 for all x E E, i.e., the operator A is bounded below. This boundedness implies that A has a trivial kernel and a closed range. If, in addition, A~ ----; A* strongly, then the same arguments yield that A* has a trivial kernel, too. Since clos 1mB =.L (Ker B*) := {x E E: f(x) = 0 for all f E Ker B*} for every bounded linear operator B, the range of A is all of E. Thus, A is invertible. The strong convergence of A;;-l to A -1 follows from and the strong convergence of adjoint sequence follows similarly. o Compactness and strong convergence. The notions of compactness and of strong convergence are intimately related by the following theorem. Theorem 1.1.3 Let An, A E L(E). Then IIAnx - AxilE ----; 0 for every x E E if and only if IIAnK - AKIIL(E) ----; 0 for every K E K(E). Proof. Let (An) be a strongly convergent sequence with strong limit A, and let K be a compact operator, i.e., suppose the set M := {Kx : Ilxll :S I} is relatively
- 23. 1.1. Generalized compactness compact. Assume that 3 IIAnK - AKI/ = sup IIAnKx - AKxl1 = sup IIAny - Ayll IIxl19 yEM does not converge to zero as n ---> 00. Then there are an c > aas well as an infinite sequence (Yn) <:;;; M such that IIAnYn - AYnl1 > c. Since M is relatively compact, one can extract a subsequence (zn) of (Yn) which converges to a Z E E. For every n, this choice implies c < IIAnzn - Aznll < II(An - A)(zn - z)11 + II(An - A)zll < supllAk - Allllzn - zll + I/(An - A)zll· k The right-hand side of this inequality becomes smaller than any prescribed c > 0 if only n is large enough. This contradiction verifies the 'only if' part of the assertion. For the 'if'-part, given x E E, choose a functional f E E* such that Ilfll = 1 and f(x) = Ilxll, which is possible by the Hahn-Banach theorem, and consider the operator KxY := f(y)x, Y E E. (1.1) This operator is ofrank one, hence compact, and I/KxIIL(E) = Ilxl/E. Since [IAnK- AKII ---> 0 for every compact operator K by hypothesis, one has IIAnx - AxilE = I/KAnx - KAxIIL(E) = IIAnKx - AKxIIL(E) ---> a for every x E E. Hence, An ---> A strongly. 0 Consequently, the strong convergence of An to A, considered as operators on E, is equivalent to the strong convergence of An to A, considered as operators of left multiplication on K(E). In Section 1.1.4 we are going to employ this equivalence in order to introduce other kinds of strong convergence. Actually, the algebra L(E) is isometrically and isomorphically embedded into L(K(E)) via its left regular representation, which associates with every operator A E L(E) an operator Al on K(E) acting as multiplication from the left, Al : K(E) ---> K(E), K f-> AK. Proposition 1.1.4 If A E L(E), then Al E L(K(E)), and the mapping A f-> Al is an isometry. Proof. Evidently, the operator Al acts linearly on K(E), and the estimate IIAKII ::; IIAIIIIKII shows that Al is bounded and that IIAtlIL(K(E)) ::; IIAIIL(E)' For the reverse estimate, let K x be as in (1.1). Then IIAtlIL(K(E)) = IIAKIIL(E) > IIAKxIIL(E) sup sup KEK(E){O} IIKIIL(E) - xEE{O} IIKxl/L(E) IIKAxIIL(E) I/AxIIE _ IIAII XES~~O} I/Kxl/L(E) XES:~O} ~ - L(E), which proves the assertion. o
- 24. 4 Chapter 1. Limit Operators Fredhohn operators. An operator A E L(E) is called Fredholm if both its kernel Ker A and its cokernel Coker A := E lIm A have finite dimension. In this case, the range of A is closed, and the integer ind A := dim Ker A - dim Coker A is called the index of A. Let Fred (E) stand for the set of the Fredholm operators on E. Theorem 1.1.5 (a) If A is Fredholm and K is compact, then A+K is Fredholm, andind (A+K) = indA. (b) Fred (E) is open in L(E), and the function ind : Fred (E) -+ Z is continuous. (c) Fred (E) is a semi-group under multiplication, and ind is a homomorphism from Fred (E) into the additive group Z. (d) If A is Fredholm, then A* is Fredholm, and ind A* = - ind A. Compactness and Fredhohnness. The ideal of the compact operators determines the set of the Fredholm operators in the following sense. Theorem 1.1.6 An operator A E L(E) is Fredholm if and only if the coset A+K(E) is invertible in the quotient algebra L(E)jK(E), the Calkin algebra of E. The norm and the spectrum of the coset A +K(E) in L(E)jK(E) are referred to as the essential norm IIAIless and the essential spectrum O"ess(A) of A, respectively. The proofs of the preceding theorems can be found in standard textbooks on Functional Analysis. Immediate consequences of Theorem 1.1.6 are the invariance of the Fredholm property under small and under compact perturbations as well as the semi-group property of Fred (E). 1.1.2 P-compactness Let E be a Banach space, and let P = (Pn)~=o be a bounded sequence of operators in L(E). We call P an increasing approximate projection if, for every mEN, there is an N(m) EN such that (1.2) and P is a decreasing approximate projection if, for every mEN, there is an N(m) EN such that (1.3) If P = (Pn ) is an increasing approximate projection, then the sequence (Qn) with Qn := I - Pn forms a decreasing approximate projection which we call associated to P.
- 25. 1.1. Generalized compactness 5 An (increasing or decreasing) approximate projection (Pn ) is said to be proper if Pn =I- 0 and Pn =I- I for all n. If (Pn) is a proper increasing approximate projec- tion, then IlPnII 2:: 1 for all sufficiently large n. It is also clear that P* := (P;:) is an increasing (decreasing) approximate projection in L(E*) whenever P = (Pn ) is an increasing (decreasing) approximate projection in L(E). Further, every infinite subsequence of an increasing (decreas- ing) approximate projection is an increasing (decreasing) approximate projection again. In what follows, we will mainly be concerned with proper and increasing approximate projections, which therefore will be simply called approximate pro- jections. P-compactness. Every approximate projection P = (Pn ) gives rise to a substitute of the ideal of the compact operators. Definition 1.1.7 An operator K E L(E) is P-compact if IIKPn - KII -> 0 and IlPnK - KII -> 0 as n -> 00. By K(E, P) we denote the set of all P-compact operators on E, and by L(E, P) the set of all operators A E L(E) for which both AK and K A are P-compact whenever K is P-compact. It is immediate from this definition that all operators in Pare P-compact and that K(E*, P*) = {K* : K E K(E, P)}, L(E*, P*) = {A* : A E L(E, PH. (1.4) Proposition 1.1.8 Let P = (Pn) be an approximate projection and Qn := 1- Pn . (a) An operator A E L(E) belongs to L(E, P) if and only if, for every kEN, (b) L(E, P) is a closed subalgebra of L(E) which contains the identity operator, and K(E, P) is a closed ideal of L(E, P). Proof. (a) The conditions (1.5) are dearly necessary for A E L(E, P). Let, con- versely, A satisfy (1.5), and let K E K(E, P). Given c > 0, choose r such that 11K - PrKl1 < c, and choose N such that IIQnAPr II < c for all n 2:: N. Then for all n 2:: N. The other conditions can be checked similarly.
- 26. 6 Chapter 1. Limit Operators (b) It is immediate from the definitions that L(E, P) is a subalgebra of L(E) and that K(E, P) is contained in L(E, P) and forms an ideal of this algebra. To get the closedness of K(E, P), let Km be P-compact and IIKm - KII ----., O. Choose r such that 11K - Krll sup IIQnl1 < E/2, and N such that IIKrQnl1 < E/2 for all n 2:: N. Then for all n 2:: N. The 'dual' assertion IIQnKl1 ----., 0 can be checked analogously. Let now (Am) be a sequence in L(E, P) which converges in the norm to A E L(E). If K is P-compact, then (AmK) and (KAm) are sequences in K(E, P) which converge in the norm to AK and K A, respectively. Since K(E, P) is closed, one has AK, K A E K(E, P) and, hence, A E L(E, P). 0 It is also easy to see that K(E, P) is the smallest closed ideal of L(E, P) which contains the operators Pm constituting P. Invertibility in L(E, P). A delicate question is that of the inverse closedness of L(E, P) in L(E). Recall that a (not necessarily closed) subalgebra B of a Banach algebra A with identity is called inverse closed in A if whenever an element b E B is invertible in A, its inverse b-l already belongs to B. In case A is a C*-algebra with identity, every symmetric and closed subalgebra of A which contains the identity is inverse closed. For the inverse closedness of L(E, P) in L(E) we need some stronger proper- ties of P. Given an approximate projection P = (Pn);:"=o in L(E), we set So := Po and Sn := Pn - Pn- l for n 2:: 1. Further, for every bounded subset U of lR, we define Pu := LkENnu Sk and Qu := 1- Pu· (Thus, Pk has still the same meaning as above, whereas P{k} = Sk') The approximate projection P is called uniform if sup IIPul1 < 00, the supremum over all bounded U <:: R (1.6) Theorem 1.1.9 If P is a uniform approximate projection, then L(E, P) is an in- verse closed subalgebra of L(E). Proof. Let again (Qn) stand for the approximate projection which is associated to P. Further we will write m « n if PkQn = QnPk = 0 for all k :s m, and we denote the supremum in (1.6) by C. Let the operator A E L(E, P) be invertible in L(E). By (1.5), what we have to show is that Given E > 0, choose and fix a positive integer mwith IIA-11121IAII/m < E, and choose integers 0 - (1) < (1) < (1) < (1) < (2) < ... < (m-l) < (m) < (m) < (m) < (m) -rl r2 r3 r4 r l r4 r l r2 r3 r4
- 27. 1.1. Generalized compactness such that k + r(i) « k + r(il and k + r Ul « k + r(j+1) 1 1+1 4 1 for all 1 ::; I ::; 3, 1 ::; i ::; m and 1 ::; j ::; m - 1 and such that 7 IIPk+ri)AQk+r~i)11 < E/IIA-1 11 2 , (1.7) IIQk+rii)APk+r~i)11 < E/IIA-1 11 2 , (1.8) IIPk+rii)AQk+r~i) II < E/IIA-1 11 2 , (1.9) IIQk+ri+l)APk+r~i)11 < E/IIA-1 11 2 . (1.10) That is, given riil we choose r~il > riil such that k +r~i) » k +riil and that (1.7) holds, then r~i) > r~i) such that k +r~il » k +r~i) and that (1.8) is satisfied, then r~i) > r~i) which fulfills k + r~il » k + r~il and (1.9), and finally riHll > r~il such that k + riHll » k + r~il and that (1.10) is valid. Let n» k + r~ml. We set U· '= (k + r(il k + r(ilj V; '= (k + r(il k + r(ilj t· l ' 3' t· 2' 4' TT' ' - [0 k (il] U' '- [0 k (il] vi'- , + r2 , i ' - ' + r3 . Then, since n » k + r~ml » k + r~il for all i, Pk Pv/A- 1 Qn PkA-1Pu~APv,A-1Qn + PkA-1Qu,APv,A-1Qn (1.11) t '( t t with IlPkA-1Qu:APv/A-1Qnll < C(C + 1) IIA- 1 11 2 1IQu:APv/ II C(C + 1) IIA-11121IQk+rii)APk+r~i) II < C(C + I)E (1.12) due to (1.8) (observe that IIQul1 = III - Pull::; 1+C). Further, since n» k+r~il, PkA-1Pu:APv/A- 1Qn - PkA-1 Pu,AQv,A-1Qn t t - PkA-1Pui AQv/A-1Qn - PkA-1PU:Ui AQV/A-1Qn = - PkA- I PUiAPviA-IQn - PkA-1PUiAQk+r~i)A-1Qn - PkA- I Pu'uAPvA-IQn - PkA- I PU'UAQk+ (i)A-IQn t t t i t T 4 = - PkA- I PUiAPViA-1Qn - PkA-I PU:AQk+r~i)A-IQn (1.13) - PkA- I PU:UiAPviA-IQn'
- 28. 8 Chapter 1. Limit Operators For the middle term on the right-hand side of (1.13), we have by (1.9), IIPkA-1 PU,AQk+ (i)A-1Qnll • r. :S C(C + 1) IIA-11121IPk+r~i)AQk+r~i) II :S C(C + 1)c: and, for the last term, due to (1.7), IlPkA - 1 PU:UiAPViA-IQnll < IIPkA- 1 Pk+r;i) AQk+r;i)Pk+r~i)A-IQnll < C 2 (C + 1) IIA-11121IPk+r;i)AQk+r;i) II < C2 (C + 1)c: where we used that PVi = Pek+r;i),k+r~i)l = Qk+r;i) - Qk+rii) = Qk+r;i) - Qk+r;i) Qk+rii) = Qk+r;i) Pk+rii). From (1.11)-(1.15) we conclude that PkA-1Qn = -PkA-1 PUiAPviA-IQn + Di (1.14) (1.15) (1.16) where Di is an operator with norm less than CE: with c being a constant independent of i and c:. Summarizing the identities (1.16) we get m m mPkA-1Qn = - 2::PkA-IPuiAPviA-IQn + 2:: D i i=l i=l m -PkA-1Pu,u...uu",APv,u...uv",A-1Qn + 2:: D i i=l m 2:: PkA- 1 PuiAPev,u "UV",JviA-1Qn i=l since Ui n Uj = Vi n Vj = 0 for i 1:- j. For the first item in (1.17) we find IIPkA-1PU,u...UUm APv,u...uv",A-1Qn II :S C3 (C + 1) IIA-11121IAII, and for every term in the last sum in (1.17) we obtain IIPkA- 1 PUi APev,u"'UV",)ViA-1Qn II :S C(C + 1) IIA-11121IPuiAPev,u...uV",)v.ll. Since (1.17) (1.18)
- 29. 1.1. Generalized compactness (recall that k + rl « k + r~i)) and, similarly, as well as p(V,U···UVi-2) + P(k+r~i-1J, k+r~i-1JJ + P(k+r~i+1), k+r~i+1)J + P(Vi+2U ··UV",) Pk+r~i-1JP(VIU···UVi_2) + Pk+r~i-1J Qk+r~i-l) + Qk+r~i+1) Pk+r~i+1) + Qk+r~i+1J P(Vi+2U ··UV",), we can further estimate the right-hand side of (1.18) by IIPUiAP(V,U ...UV",)Vi II < IlPuiAPk+r~i-1) 1I11P(V1u...uvi - 2 ) + Qk+r~i-1) II + IlPuiAQk+r~i+1) 1IIIPk+r~i+1J + P(V,+2U...UV",) II < C(l + 2C) IIQk+r~i)APk+r~i-1JII + 2C(C + 1) IIPk+r~i)AQk+r~i+1) II < C(l + 2C) IIQk+r~i)Qk+riiJAPk+r~i-1)II +2C(C+1)IIPk+ (iJAQk+riQk+ (H1JII ::; de: T 3 4 T 2 9 with a constant d independent of i and e: due to (1.9) and (1.10). Inserting these estimates into (1.17) and dividing by m, we arrive at < 2-(C3 (C + 1) IIA-I II2 1IAII + mcc + mde:) m < (C3 (C + 1) + c + d)e: for all n» k+ri m ). Thus, IIPkA-IQnll ~ 0, and the dual assertion IIQnA-I Pkll ~ ocan be checked analogously. 0 Equivalent approximate projections. We call the approximate projections P = (Pn) and pI = (P~) equivalent if for all m 2: O. For example, every infinite subsequence of an approximate projection p is equivalent to P. Lemma 1.1.10 The approximate projections P and pI are equivalent if and only if K(E, P) = K(E, PI).
- 30. 10 Chapter 1. Limit Operators Proof. Let P = (Pn), pI = (P~), and set Qn := 1- Pn and Q~ := I - P~. Further let K E K(E, P). Then, for all m, n ::::: 0, KQ~ = (K - KPm)Q~ + KPmQ~. Given € > 0, choose m such that 11K - KPml1 < € and N(m) such that IlPmQ~11 = IIPm - PmP~11 < € whenever n ::::: N(m). Then IIKQ~II ::; €SUp IIQ~II + €IIKII for all n ::::: N(m), n whence IIKQ~11 -+ O. Similarly one gets IIQ~KII -+ 0, i.e., K E K(E, PI). Conversely, the equality K(E, P) = K(E, PI) implies that Pm E K(E, PI) for all m, i.e., Changing the roles of P and pI, we get the second condition for the equivalence of approximate projections. 0 Consequently, if P and pI are equivalent, then L(E, P) = L(E, PI). 1.1.3 P-Fredholmness Let again P = (Pn ) be an approximate projection on the Banach space E, and set Qn = 1- Pn. The following definition is motivated by Theorem 1.1.6. Definition 1.1.11 An operator A E L(E, P) is called P-Fredholm if the coset A + K(E, P) is invertible in the quotient algebra L(E, P)jK(E, P). This definition implies that the P-Fredholmness of an operator is invariant both under sufficiently small and under P-compact perturbations, and that the product of P-Fredholm operators is P-Fredholm again. It is also clear that if P and P' are equivalent approximate projections, then an operator is P-Fredholm if and only if it is pI-Fredholm. We call the norm and the spectrum of the coset A + K(E, P) in L(E, P)jK(E, P) the P-essential norm and the P-essential spectrum of A. Proposition 1.1.12 An operator A E L(E, P) is P-Fredholm if and only if there exist operators C, D E L(E, P) and an mEN such that QmAC = Qm and DAQm = Qm' (1.19) Operators A E L(E, P) which satisfy (1.19) are also called invertible at infinity. Proof. If A E L(E, P) is P-Fredholm, then there exist operators C E L(E, P) and K E K(E, P) such that AC = 1+ K. Multiplying this equality by Qr from the left-hand side and adding Pr to both sides yield QrAC + Pr = 1+ QrK. Choose
- 31. 1.1. Generalized compactness 11 r such that IIQrKl1 < 1 and m such that QmPr = 0 and, hence, QmQr = Qm (which is possible due to (1.2)). Then 1+ QrK is invertible, and we get QrAC(I + QrK)-l + Pr(I + QrK)-l = I. Multiplying this equality by Qm from the left-hand side yields the invertibility at infinity of A from the right-hand side. (Observe that the operator C(I +QrK)-l lies in L(E, P) by Neumann series.) The invertibility at infinity from the left-hand side follows analogously. Conversely, (1.19) implies AC = I - Pm + PmAC =: 1+ K 1 and DA = I - Pm + DAPm =; 1+ K2 with K 1, K2 P-compact. Hence, A is P-Fredholm. 0 1.1.4 P-strong convergence Let P = (Pn)~=o c L(E) be an approximate projection. Theorem 1.1.3 suggests the following definition. Definition 1.1.13 Let An E L(E, P). The sequence (An) converges P-strongly to A E L(E) if, for all K E K(E, P), both II(An - A)KIIL(E) ----> 0 and IIK(An - A)IIL(E) ----> o. In this case we write An ----> A P-strongly or A = P-limAn. The P-strong convergence of bounded sequences can be characterized as follows. Proposition 1.1.14 If (An) is a bounded sequence in L(E, P), then (An) converges P-strongly to A E L(E) if and only if II(An - A)Pmll ----> 0 and IlPm(An - A)II ----> 0 for every fixed Pm E P. (1.20) Proof. Since P <:;;; K(E, P), the P-strong convergence of An to A implies (1.20). Conversely, let (1.20) be satisfied for operators An and A which are uniformly bounded, and let K be P-compact. Then, for every Pm, II(An - A)KII :::; II(An - A)PmKII + II(An - A)(I - Pm)KII :::; II(An - A)PmIIIIKII + IIAn - All II(I - Pm)KII· The right-hand side of this estimate becomes as small as desired if m is large enough. Thus, II(An - A)KII ----> 0 for every K, and the dual condition follows analogously. 0 For example, it is immediate from (1.2) that Pn ----> I P-strongly. On the other hand, (1.20) indicates that the notion of P-strong convergence has some serious defects. For example, the P-strong limit in not unique in general (choose P as the constant sequence (P) with a non-trivial projection P). So we will have to impose further conditions on P which guarantee, for example, the uniqueness of the P-strong limit.
- 32. 12 Chapter 1. Limit Operators Approximate identities. The approximate projection P = (Pn ) is called an ap- proximate identity if sup IlPnxll ~ Ilxll for each x EE, n and it is called a symmetric approximate identity if, besides (1.21), sup IIP~/II ~ IIIII for each IE E*. n (1.21) For A E L(E, P), consider the operators of left and right multiplication Al : K f---* AK and Ar : K f---* K A on the Banach space K(E, P). The following proposition is the P-analogue of Proposition 1.1.4. It shows that L(E, P) is topologically embedded into the Ba- nach algebra L(K(E, P)) under each of the mappings A f---* Al (= left regular representation) and A f---* Ar (= right regular representation). As a consequence we get that the P-strong convergence of An to A is essentially equivalent to the (usual) strong convergence of the operators (An)1 and (An)r to Al and Ar on K(E, P) , respectively. Proposition 1.1.15 Let P = (Pn) S;; L(E) be an approximate identity, A E L(E, P), and set Cp := sup IlPn II. Then (1.22) and IIAdIL(K(E,P)) :::::: IIAIIL(E) :::::: CfIIAdIL(K(E,P))' (1.23) Proof. The first inequality in (1.22) is obvious. For the second one, let A E L(E, P) and c > O. Choose an Xo EE with Ilxoll = 1 such that IIAxol1 ~ IIAII - c, and let Pn E P be such that IlPnAxol1 ~ IIAxoll - c, which is possible due to (1.21). Then, since P C K(E, P), sup IIKAII > IlPnA11 > IlPnAxol1 > _1 (IIAII- 2c). KEK(E, P){O} IIKII - IlPnll - Cp - Cp Thus, IIAII :::::: Cp IIArll· The first inequality in (1.23) is again obvious. Let c > O. As we have just seen, there is an m such that 1 IlPmA11 ~ IlPmAII/llPmll ~ C p IIAII - c. Choose n such that IlPmAQn11 = IlPmA-PmAPnll < c (see Proposition 1.1.8 (a)). Then 2 IIAPnl1 1 Cp~ ~ CpllAPnl1 ~ IIPmAPnII ~ IlPmAII-llPmAQnll ~ Cp IIAII- 2c. Hence, IIAII :::::: Cf IIAdIL(K(E,P))' 0
- 33. 1.1. Generalized compactness 13 Corollary 1.1.16 Let P be an approximate identity. Then no sequence in L(E, P) possesses more than one P -strong limit. Indeed, if IIK(An - A)II ---+ 0 and IIK(An - B)II ---+ 0 for all P-compact operators K, then IIK(A - B)II = 0 for all K E K(E, P). Hence, (A - B)r = 0, whence A - B = 0 by the preceding proposition. 0 Here are some properties of P-convergent sequences which are more or less imme- diate consequences of Propositions 1.1.2 and 1.1.15. In particular the hypotheses in assertion (c) are chosen such that the invertibility of the P-strong limit follows without effort from Proposition 1.1.2. We will discuss this invertibility problem under weaker assumptions for P in Section 1.1.5. Proposition 1.1.17 Let P be an approximate identity, and let (An) and (Bn) be sequences of operators in L(E, P) which converge P-strongly to operators A, BE L(E), respectively. (a) The operator A belongs to L(E, P), the sequence (An) is uniformly bounded, and IIAIIL(E) :s Cp liminf IIAnIIL(E)' In particular, L(E, P) is a closed subspace of L(E) with respect to P-strong convergence. (b) An + Bn ---+ A + Band AnBn ---+ AB P-strongly. (c) If P is uniform, if the Pn converge *-strongly to the identity operator, and if the operators An are invertible for all sufficiently large n and the norms of their inverses are uniformly bounded, then A is invertible, and A;;-1 ---+ A-1 P -strongly. (d) If P is symmetric, then A~ ---+ A * P* -strongly. Proof. Let An ---+ A P-strongly and K E K(E, P). Then we have AnK E K(E, P) and K An E K(E, P) since, by assumption, An E L(E, P). Further, IIAnK - AKII ---+ 0 and 11KAn - K All ---+ O. Since K(E, P) is closed, this implies that AK and K A belong to K(E, P), i.e., A is in L(E, P). Consequently, the convergence II(An - A)KII ---+ 0 for all K E K(E, P) is equivalent to the strong convergence of the operators An to A on K(E, P) if we identify operators on E with their action on K(E, P) as left multiplication operators. Similarly, the condition IIK(An - A) II ---+ 0 for all K E K(E, P) is equivalent to the strong convergence of the operators An to A on K(E, P), but now considered as acting as right multiplication operators on this Banach space. This shows that P-convergence is essentially strong convergence on a specified Banach space. Thus, by the Banach-Steinhaus theorem, the sequence ((An)r) of right multiplication operators is uniformly bounded, and IIArl1 :s liminf II(An)rll. Taking into account (1.22) we arrive at assertion (a). Similarly, (b) follows from Proposition 1.1.2.
- 34. 14 Chapter 1. Limit Operators To prove (c), notice that the invertibility of A follows from Proposition 1.1.2 (c). By Theorem 1.1.9, the inverse of A belongs to L(E, P). Thus, for every K E K(E, P), we have A-IK E K(E, P) and consequently, II(A- I - A~I)KII ::; IIA~IIIII(An - A)A-I KII ::; CII(An - A)A-I KII ---., O. The dual condition follows similarly, and assertion (d) is obvious. o Approximate identities with specific properties. In what follows we will often meet approximate projections P = (Pn ) with sup IlPnxll = Ilxll for every x E E. n (1.24) Evidently, every approximate projection which satisfies this condition is an ap- proximate identity. Condition (1.24) implies that IlPnll ::; 1 for all n. Together with (1.2), this shows that IIPnl1 = 1 for all sufficiently large n and that Cp = 1. Thus, under the assumption (1.24), the inequalities in (1.22) and (1.23) become equalities. Moreover, the limit limn--->oo IlPnxll exists and lim IlPnxll = Ilxll for every x E E. n--->oo (1.25) Indeed, let x E E and E > O. Choose m such that Ilxll - E ::; IlPmxll and N such that IlPmxll = IlPmPnxl1 ::; IlPnxll for all n 2:: N. Thus, Ilxll - E ::; IlPnxll ::; Ilxll for all sufficiently large n. Moreover, an obvious modification of the proof of Proposition 1.1.15 under the additional assumption (1.24) also yields that and lim IlPnA11 = IIAII for all A E L(E) lim IIAPnll = IIAII for all A E L(E, P). (1.26) (1.27) Another class of approximate identities is provided by approximate projections which converge strongly to the identity operator. (It is again obvious that the strong convergence Pn ---., I forces condition (1.21).) We will call an approximate identity P = (Pn) perfect if both Pn ---., I and P~ ---., 1* strongly. Perfect approxi- mate identities are symmetric. If P is a perfect approximate identity, then Theorem 1.1.3 implies that K(E) ~ K(E, P). This shows that Fredholm operators are P-Fredholm, and that P-strong convergence implies common strong convergence. For the latter, notice that P-strong convergence of (An) to A implies IIAnKx - AKxl1 ---., 0 where K x is as in (1.1), whence IIAnx - Axil ---., 0 for every x E E. Conversely, if all operators Pn are compact in the common sense, then every operator K E K(E, P) is the uniform limit of the compact operators PnK, whence K(E, P) ~ K(E). Thus, if P is perfect and P C K(E), then K(E, P) = K(E) and L(E, P) = L(E), and all 'P-notions' reduce to their common meaning.
- 35. 1.1. Generalized compactness 15 (1.29) 1.1.5 Invertibility of P-strong limits In order to get information about the invertibility of the limit of a P-strongly convergent sequence without assuming the symmetry or the strong convergence of the approximate identity P, we have to restrict the class of sequences under consideration. To have a frame in which we can work, let F(E) stand for the set of all bounded sequences (Ank:~o of operators in L(E). Provided with the operations (An) + (Bn) := (An + Bn), (An)(Bn):= (AnBn), a(An):= (aAn) and with the norm II(An)11 := sup IIAnll, this set becomes a Banach algebra with identity element (1). If E is a Hilbert space, then the involution (An)' := (A~) makes F(E) even to a C*-algebra. To motivate the following definition, recall that an operator A E L(E) belongs to L(E, P) if, for every m 2:: 0, IlPmAQn11 -+ ° and IIQnAPml1 -+ ° as n -+ 00. (1.28) Definition 1.1.18 A sequence (An) E F(E) belongs to the class F(E, P) if, for every m 2:: 0, sup IlPmAkQnl1 -+ ° and sup IIQnAkPmll -+ ° as n -+ 00. k~O k~O It follows immediately from (1.28) that if (An) E F(E, P), then every operator An belongs to L(E, P) and that, conversely, for every operator A E L(E, P), the constant sequence (A) belongs to F(E, P). It is also evident that the sequence (Pn) is in F(E, P). Theorem 1.1.19 F(E, P) is a closed subalgebra of F(E). If the approximate iden- tity P is uniform, then F(E, P) is inverse closed in F(E). Proof. It is obvious that F(E, P) is a linear space. We show that the product of two sequences (An), (Bn) E F(E, P) also belongs to F(E, P). For m, r 2:: 0, we have < sup IlPmAkPrBkQnl1 + sup IlPmAkQrBkQnl1 k k < C sup IlPrBkQnl1 + C sup IlPmAkQrll. k k We choose and fix an r such that the second term in (1.29) becomes as small as desired, and then an no such that the first term in (1.29) becomes smaller than a given constant for all n 2:: no. This shows that F(E, P) is an algebra, and the proof of its closedness is also fairly standard. The inverse closedness of F(E, P) in F(E) can be verified in exactly the same way as the inverse closedness of L(E, P) in L(E) in Theorem 1.1.9. 0 Let Eo stand for the closure in E of UmIm Pm.
- 36. 16 Chapter 1. Limit Operators Lemma 1.1.20 (a) Eo consists of all x E E with Qnx --+ O. (b) Eo is a closed linear subspace of E. (c) The subspace Eo is invariant for operators in L(E, P). Proof. Let x E Eo and € > O. Choose mEN and y E E such that IIx - Pmyll < €, and choose N such that QnPm = PmQn = 0 for all n 2: N. Thus, for all n 2: N, yielding Qnx --+ O. Conversely, Qnx --+ 0 for some x E E implies x = lim Pnx, i.e., x E Eo. Assertion (b) follows immediately from (a). For (c), let A E L(E, P) and x E E be such that Qnx --+ O. Then, for all m, where the right-hand side becomes less than any € > 0 if m is chosen such that IIQmxl1 < €/2 and if n is sufficiently large. Thus, QnAx --+ 0 and Ax E Eo. D Here is the desired generalization of Proposition 1.1.17 (c). Proposition 1.1.21 Let P be a uniform approximate identity, and let (An) be a sequence in F(E, P) with P-strong limit A. If all operators An are invertible, and if the norms of their inverses are uniformly bounded, then AIEo is invertible, and A;;-llEo --+ (AIEo)-l strongly. Proof. From Pmx = A;;-lAnPmx we conclude that IIPmxl1 :::; CIIAnPmxl1 for all x E E and m 2: O. Passage to the limit as n --+ 00 yields IlPmxll :::; CIIAPmxll, and letting m go to infinity in case x E Eo shows that [Ixll :::; C[[Axll for all x E Eo. (1.30) From Proposition 1.1.17 we know that A E L(E, P). Thus, by Lemma 1.1.20, Eo is an invariant subspace for A, and from (1.30) we conclude that the operator AIEo has a trivial kernel and a closed range. We claim that the range of AIEo is all of Eo. For every k, m 2: 0, we have 11(1 - AA;;-l)Pkll II(An - A)A;;-lPkll II(An - A)(Pm + Qm)A~l Pkll < CII(An - A)Pmll + CIIQmA~lPkll. Given € > 0, choose and fix m such that IIQmA;;-lPkll < € uniformly with respect to n which can be done since the sequence (A;;-l) belongs to F(E) and, hence, to F(E, P) by Theorem 1.1.19. Further, choose no such that II(An - A)Pmll < € for all n 2: no· These choices guarantee that II(I - AA;;-l )PkII :::; 2C€ for all n 2: no, whence (1.31)
- 37. 1.2. Limit operators 17 as n ---> 00. Since V is uniform, the operators A;:;-l belong to L(E, V), hence, they leave the space Eo invariant. Thus, (1.31) shows that Pmx is the norm limit of vectors in ImAIEo' Since we already know that this range is closed, we obtain Pmx E ImAIEo for every x E E and m ::::: o. Employing the closedness of AEo once more we get ImAIEo = Eo whence the invertibility of AIEo' Finally, it follows from 11(1 - AA;:;-l)Pkll---> 0 that whence the strong convergence of A;:;-llEo to (AIEo)-l. o If, in particular, Pn ---> I strongly, then the preceding proposition guarantees the invertibility of A = V-limAn on E = Eo without further symmetry assumptions on V (but only for operators (An) in F(E, V), whereas Proposition 1.1.17 (c) gives this result for sequences in F(E)). Let us also mention that in Chapters 2 and 6 we will meet situations where Eo is a proper subspace of E, but where E can be identified with the second dual of Eo, which also can be used to prove the invertibility of the V-strong limits on E, not only on Eo. 1.2 Limit operators The theory of limit operators which we are going to introduce in this section will provide us with an adequate tool to investigate band and band-dominated operators. 1.2.1 Limit operators and the operator spectrum Let N be a positive integer. We suppose that the additive group TLN acts contin- uously on the Banach space E, i.e., that there is a bounded family V = {VdkEiZN of operators Vk E L(E) such that VkVi = Vk+l for all k, l E TLN and Vo = I. It is possible to formulate and prove many of the following notions and results also for more general (in particular, non-commutative) locally compact topological groups in place of 7LN . We renounce to consider these more general settings here since they would not only blow up the size of the book essentially, but would also be redundant for most of the applications we have in mind. Only in Chapters 4 and 7, we will have to deal with actions of discrete Heisenberg groups in place of 7LN . In these cases, the analogues of the following notions and results are evident. Let 1{ stand for the set of all sequences h : N ---> TLN which tend to infinity in the sense that, for every R > 0, there is an mo such that Ih(m)1 > R for all m ::::: mo. Further we assume that there is an approximate identity V = (Pm) on E which is related with the group action V as follows. For every m, n E N, there is an R > 0 such that PmVkPn = 0 for alllki > R, (1.32)
- 38. 18 and for every mEN and k E 7LN , Chapter 1. Limit Operators there is an no EN such that PmVkQn = QnVkPm = 0 for all n 2: no. (1.33) The latter condition ensures that V ~ L(E, P) due to Proposition 1.1.8 (a). Definition 1.2.1 Let A E L(E, P), and let hE H. The operator Ah E L(E) is called limit operator of A with respect to h if (1.34) The set O"op(A) of all limit operators of A is called the operator spectrumof A. An operator can possess only one limit operator with respect to a given sequence (Corollary 1.1.16) which justifies the notation Ah for the limit operator. Observe also that if9 is a subsequence of h E H, then 9 is also in H, and if the limit operator Ah exists for an operator A E L(E, P), then Ag also exists, and Ag = A h. Let us further recall from Proposition 1.1.14 that Ah is the limit operator of A with respect to the sequence h if and only if and for every Pm E P. Here are some elementary properties of limit operators which are immediate consequences of Proposition 1.1.17. Proposition 1.2.2 Let P be an approximate identity, let h E H, and let A, B be operators in L(E, P) for which the limit operators Ah and Bh exist. Then (a) IIAhll::::: e11A11 with a constant e independent of A and h. (b) the limit operators (A + B)h and (AB)h exist and (A + B)h = A h + B h and (AB)h = AhBh. (c) if P is perfect and A is invertible, then Ah is invertible, the limit operator (A-1)h exists, and (A-l)h = (Ah)-l. (d) if P is symmetric, then the limit operator (A*)h (taken with respect to P*) exists and (A*)h = (Ah)*. (e) if e, e(m) E L(E, P) are operators with lie - e(m)11 -+ 0, and if the limit operators(e(m)) h exist for all sufficiently large m, then the limit operator eh exists, and Ileh - (e(m))hll -+ O. Proposition 1.2.3 The operator spectrum of an operator in L(E, P) is bounded and closed with respect to P-strong convergence.
- 39. 1.2. Limit operators 19 Proof. The boundedness of the operator spectrum follows from Proposition 1.2.2 (a). For the proof of its closedness, let AE L(E) be the P-strong limit of a sequence (A(k))~l of limit operators of an operator A E L(E, P). Due to the definition of a limit operator, there exists an increasing sequence of points h(I), h(2), ... E'lIP such that IIW-h(k)AVh(k) - A(k)).Pz11 < l/k for all 1 = 1, ... , k. and II.PzW-h(k)AVh(k) - A(k))11 < l/k for all 1 = 1, ... , k. We claim that Ais the limit operator of A with respect to the sequence (h(k))~l' Let Pm E P and c > O. Choose k1 such that II(A(k) - A)Pmll < c/2 for all k ::::: k1. and k2 such that l/k2 < c/2. Then, for all k ::::: max{m, k1 , k2 }, The 'dual' condition follows analogously. This proves our claim. o If B is a limit operator of an operator A then V_kBVk is also a limit operator of A for every k E ZN. Indeed, let h be a sequence in H such that B = Ah . Then P-strongly as n --+ 00. Thus, operator spectra are shift invariant. In combination with the preceding proposition, this has the following nice consequence. Corollary 1.2.4 Every limit operator of a limit operator of A is a limit operator of A. 1.2.2 Operators with rich operator spectrum Operators A E L(E, P) which possess limit operators Ah for sufficiently many sequences h are of particular interest. Definition 1.2.5 An operator A E L(E, P) is called an operator with rich operator spectrum or simply a rich operator if every sequence h E H contains an infinite subsequence 9 such that the limit operator of A with respect to 9 exists. The set of all operators with rich spectrum will be denoted by L$(E, P) . For example, the operators Vk are rich, and aop(Vk) = {Vd for all k E ZN. The richness of P-compact operators is part of the following proposition. Moreover, assertion (c) of that proposition states that richness is essentially a sequential compactness condition with respect to P-convergence.
- 40. 20 Chapter 1. Limit Operators Proposition 1.2.6 Let P be an approximate identity. (a) L$(E, P) is a closed subalgebra of L(E, P). (b) K(E, P) is a closed ideal of L$(E, P). In particular, lJop(K) = {O} for every P -compact operator K. (c) An operator A E L(E, P) is rich if and only if the set {V_kAVdkEZN of its shifts is relatively sequentially compact with respect to the P -strong conver- gence. Proof. (a) Let A, B E L$(E, P) and h E H. By hypothesis, there are subsequences f of hand 9 of f such that Af and Bg exist. From Proposition 1.1.12 (b) we know that then (A + B) and (AB)g exist, hence, L$(E, P) is an algebra. Now let (A(k})k'=l be a sequence in L$(E, P) which tends uniformly to an operator A E L(E, P), and let hE H. By hypothesis, we can find a subsequence gl of h such that A~~) exists, then a subsequence g2 of gl such that A~;) exists, and so on. Proceeding in this way, we obtain for each k ~ 2 a subsequence gk of gk-1 such that A~:) exists. Define a new sequence 9 by g(k) := gk(k). Evidently, 9 is a subsequence of h, and the limit operators A~m) exist for all m. Then, by Proposition 1.2.2 (e), the limit operator Ag exists, too. Thus, A E L$(E, P), showing the closedness of L$(E, P) in L(E, P). (b) Our next objective is the inclusion K(E, P) c L$(E, P). Let h E Hand Pm E P. Then, for every K E K(E, P) and Pn E P, IIV-h(k)KVh(k)Pmll < IIV-h(k)KPnVh(k) Pm II + IIV-h(k)KQnVh(k)Pmll < CllPnV,'(k) Pm II + C1IKQnll with a constant C (recall the uniform boundedness of the families P and V). Given c: > 0, choose n such that CllKQnl1 < c: and then ko such that CllPnVh(k) Pm II < c: for all k ~ ko (compare (1.32)). Then IIV-h(k) KVh(k) Pm II :::; 2c:C for all k ~ ko. The convergence IlPmV-h(k)KVh(k) II -+ 0 follows analogously. Hence, K(E, P) c L$(E, P), and the operator spectrum of a P-compact operator is the singleton {O}. (c) Let (V-h(n)AVh(n))nEN be a sequence in {V-kAVdkEZN. If the sequence h is bounded, then it possesses a constant subsequence g, and (V-g(n)AVg(n))nEN is evidently a convergent sequence. If h is unbounded, then it has a subsequence l which tends to infinity. Then, by definition of richness, there is a subsequence 9 of l such that (V-g(n)AVg(n))nEN converges P-strongly. The reverse implication is obvious. 0 Corollary 1.2.7 If A E L$(E, P), then O'op(A) is sequentially compact with respect to the P -strong convergence. Indeed, the set {V-kAVd kEZN is relatively sequentially compact with respect to the P-strong convergence by Proposition 1.2.6 (c). By definition, the operator spectrum of A is contained in the P-strong closure of this set, and the operator
- 41. 1.2. Limit operators 21 spectrum is P-strong closed by Proposition 1.2.3. Being a closed subset of a se- quentially compact set, aop(A) is sequentially compact with respect to the P-strong convergence. The perfect case. In what follows we will mainly be concerned with the P- Fredholm properties of operators in L$(E, P) and with the invertibility of their limit operators. The most complete information on invertibility of limit operators, which we have at our disposal is Proposition 1.2.2 (c), where a perfect approximate identity is assumed. So we start with focusing our attention on the perfect case. Proposition 1.2.8 Let P be a perfect approximate identity. Then L$(E, P) is in- verse closed in L(E, P). IfP is moreover uniform, then L$(E, P) is inverse closed in L(E). Proof. Let A E L$(E, P) be invertible in L(E, P), and let h E H. Then there is a subsequence 9 of h such that Ag exists. By Proposition 1.2.2 (c), the limit operator (A-1)9 also exists. Thus, A-I has rich spectrum, which reveals the inverse closedness in L(E, P). The inverse closedness in L(E) follows from Theorem 1.1.9. o Proposition 1.2.9 Let P be a perfect approximate identity. If A E L(E, P) is a P-Fredholm operator, then all limit operators of A are invertible, and the norms of their inverses are uniformly bounded. Proof. Let A E L(E, P) be a P-Fredholm operator, i.e., there exist operators D E L(E, P) and T I , T2 E K(E, P) such that DA = I + TI and AD = I + T2 . If h E 11. is a sequence such that the limit operator Ah exists, then, for every P-compact operator K, and, consequently, Passage to the limit n ---+ 00 yields IIKII :::; C1IAhKII for all K E K(E, P) (recall from Proposition 1.2.6 that (TI)h exists and is the zero operator). Analo- gously, IIKII :::; GllAhKl1 for all K E K(E*, P*). Since P is perfect, we have K(E) ~ K(E, P) as well as K(E*) ~ K(E*, P*). Thus, we can replace K by a rank one operator as in (1.1) in the latter two estimates to get Ilxll :::; G11Ahx11 and Ilfll:::; G11Ahf11 for all x E E and f E E*. Hence, Ah is invertible, and II(Ah)-III :::; G. 0
- 42. 22 Chapter 1. Limit Operators Proposition 1.2.10 Let P be a perfect and uniform approximate identity. Then L$(E, P)jK(E, P) is an inverse closed subalgebra of L(E, P)jK(E, P). Proof. Let A E L$(E, P) be a P-Fredholm operator and let D E L(E, P) be a regularizer of A (i.e., the coset D + K(E, P) is the inverse of A + K(E, P)). Since any two regularizers of A differ by a P-compact operator, and since every P- compact operator belongs to L$(E, P), it is sufficient to prove that D E L$(E, P) for at least one regularizer of A. We choose D such that DAQm = Qm for some mEN (compare Proposition 1.1.12). Now let h E 'H. Since A is rich, there is a subsequence 9 of h for which Ag exists. For every P-compact operator K we then have V_g(n)DVg(n)AgK = V_g(n)DVg(n)(Ag - V_g(n)AQmVg(n))K + V_g(n)DAQmVg(n)K = V_g(n)DVg(n)(Ag - V-g(n) AVg(n) V-g(n)QmVg(n))K + V-g(n)QmVg(n)K. The right-hand side of this equality tends to K in the operator norm (observe that IIV-g(n)QmVg(n)K - KII = IIV-g(n)PmVg(n)KII ---+ 0 by Proposition 1.2.6). Hence, V_g(n)DVg(n)Ag ---+ I P-strongly as n ---+ 00. Since Ag is invertible (Proposition 1.2.9), and since L(E, P) is inverse closed in L(E) (Theorem 1.1.9), this yields the P-strong convergence of V_g(n)DVg(n) to A;l. 0 The preceding results suggest that limit operators are closely related with Fred- holmness. One of our main objectives in this book is to single out classes of oper- ators for which the converse of Proposition 1.2.9 is true. The non-perfect case. As in Section 1.1.5, to derive the invertibility of the limit operators of an invertible operator under weak conditions for P, we have to restrict the class of operators under consideration. Definition 1.2.11 An operator A E L(E) belongs to the class L(E, P, V) if, for every m 2: 0, sup IlPmV-kAVkQnll---+ 0 and kE7L.N sup IIQnV-kAVkPmll---+O asn---+oo. kE7L.N Theorem 1.2.12 Let P be an approximate identity. Then L(E, P, V) is a closed subalgebra of L(E) which contains K(E, P) as its closed ideal. If P is uniform, then L(E, P, V) is inverse closed in L(E). Proof. The closedness and inverse closedness of L(E, P, V) can be checked as in Theorem 1.1.19 (one can also refer directly to this theorem because, after rear- ranging, the sequence (V_kAVk) can be viewed of as an element of F(E, P)).
- 43. 1.3. Algebraization 23 We are left with the implication K(E, P) c L(E, P, V). Let K E K(E, P). Since IlPsKPs - KII ~ 0 as s ~ 00, and since L(E, P, V) is dosed, it is sufficient to show that PsKPs E L(E, P, V) for every s. Let us check, for example, that for every m E No By (1.32), there is an R > 0 such that PmV-kPs = 0 for all Ikl 2: R. So it remains to show that lim IlPmV-kPsKPsVkQnll = 0 for every k E ZN with Ikl < R. n-+oo This assertion follows immediately from (1.33). o Specifying Proposition 1.1.21 to the present context essentially yields the following. Proposition 1.2.13 Let P be a uniform approximate identity. If A E L(E, P, V) is invertible and if the limit operator Ah exists for a sequence h E H, then AhlEo is invertible, (A-IIEo)h exists, and (AhIEo)-1 = (A-IIEo)h. Moreover, the norms of the inverses of the operators AhlEo are uniformly bounded. Proof. Let A E L(E, P, V) be invertible, and let h E H be a sequence for which the limit operator Ah exists. Then the sequence (V-h(n)AVh(n)) belongs to F(E, P) and is invertible in F(E). Thus, Ahl Eo is an invertible operator by Proposition 1.1.21. The P-strong convergence of V-h(n)A-IIEo Vh(n) to (AhIEo)-1 on Eo follows in the standard way: IIV-h(n)A-IVh(n)Pm - (AhIEo)-1 Pmll :s sup IlVkI12 1IA-I IIII(Ah - V-h(n)AVh(n))(AhIEo)-1 Pmll ~ 0 k and the dual assertion can be checked analogously. Finally, the uniform bounded- ness of the inverses of the limit operators is a consequence of Ilxll :s c IIAhxl1 for all x E Eo with a constant C independent of the sequence h which can be seen as in (1.30). o If, in particular, Pn ~ I strongly, then Proposition 1.2.13 yields the invertibility of Ah on E = Eo without symmetry assumptions for P (but only for operators A in L(E, P, V), whereas Proposition 1.2.9 gives this result for operators in L(E, P)). 1.3 Algebraization The algebraic properties of the mapping A 1---4 Ah are summarized in Proposition 1.2.2. It says essentially that if h E H and if 13 is a subalgebra of L(E, P) such that the limit operator Ah exists for every A E 13, then the mapping A 1---4 Ah is
- 44. 24 Chapter 1. Limit Operators an algebra homomorphism from B into L(E). But, of course, this mapping is not an algebra homomorphism on all of L$(E, P); it is not even defined on L$(E, P). In this section we will briefly discuss two ways to algebraize the concept of limit operators in order to get algebra homomorphisms. 1.3.1 Algebraization by restriction Here we will single out subalgebras of L$(E, P) on which the mapping A I----' A h acts homomorphically. Proposition 1.3.1 Let B be a separable subset of L$(E, P). Then every sequence h E H possesses a subsequence 9 such that the limit operator Ag exists for every AE B. Proof. Let {Bn}n>l be a dense subset of B, and let ho := h E H. Since the operators B n are rich, we find, for every n ~ 1, a subsequence hn of hn- 1 such that the limit operator of B n with respect to hn exists. Set g(n) := hn(n). Then 9 E H, and the limit operators (Bn)g exist for every n. If now A E Band h E H, then there is a subsequence (An) of {Bn} which tends to A in the norm, and there is a subsequence 9 of h such that all limit operators (An)g exist. By Proposition 1.2.2 (e), the limit operator A g exists, too. o For each subset B of L(E, P), we let HB stand for the set of all sequences h E H for which the limit operator Ah exists for each A E B. Proposition 1.3.2 Let B be a separable subalgebra of L$(E, P). Then the mapping A I----' A h is a continuous algebra homomorphism on B for every h E HB, and Proof. The first assertion is a consequence of Proposition 1.2.2 (and holds for any subalgebra of L(E, P)). For the second one, let A h be a limit operator of A E B. By Proposition 1.3.1, there is a subsequence g of h which belongs to HB. Thus, Ah = Ag E {Ah : h E HB}. 0 One can push this result a little bit further by including P-compact operators. Proposition 1.3.3 Let B be a separable subalgebra of L$(E, P), and let A be the smallest closed subalgebra of L(E) which contains B and the ideal K(E, P). Then every sequence h E H possesses a subsequence 9 such that the limit operator A g ex- ists for every A E B, the mapping A I----' Ah is a continuous algebra homomorphism on A for every h E HA, and
- 45. 1.3. Algebraization 25 Proof. Let A E A, and let {Bn}n~l be a dense subset of B. There are a subsequence (An) of {Bn } and a sequence (Kn) of P-compact operators such that An + K n tends to A in the norm. The operators An + K n are rich by assumption and by Proposition 1.2.6. Thus, the assertion follows as in the preceding proposition. 0 From Proposition 1.2.6 we further know that K h = 0 for every P-compact operator K and every sequence h E H. Thus, if A is as in Proposition 1.3.3 and h E HA, then the mapping Wh : AIK(E, P) ---+ L(E), A + K(E, P) I---t Ah (1.35) is well defined, and it is a continuous algebra homomorphism. This homomorphism is unital if A contains the identity operator. 1.3.2 Symbol calculus In this section we will see how to associate an operator-valued function smb A with every operator A E L$(E, P) such that the set of the values of smbA coincides with the operator spectrum O"op(A) and such that the mapping A I---t smb A is a continuous algebra homomorphism. In this case we call 8mb a symbol mapping and smb A the symbol of A. Given A E L(E, P), let HA denote the set of all sequences h E 'H such that the limit operator Ah exists. A most natural candidate for the symbol of A is the function (1.36) There arise obvious difficulties if one wants to add or to multiply such 'symbols'. Indeed, smboA and smboB are defined on 'HA and HB, respectively. Thus, their 'sum' smboA + smboB is naturally defined on HA n HB only. On the other hand, the 'symbol' smbo(A+B) of A+B is defined on HA+B which can be a much larger set than HA n HB. For example, HA n H-A = HA, whereas HA-A = Ho = H. Thus, the mapping smbo defined by (1.36) is not a symbol mapping in our sense. To avoid these difficulties, we are going to introduce an equivalence relation which will allow us to identify smboA + smbo(-A) with smboO for all A E L$(E, P). For, we first have to define an equivalence relation on 'H as follows. Two sequences g, h E H are equivalent if there exist k, LEN such that g(k + n) = h(l + n) for all n E N. (1.37) Definition 1.3.4 A subset D of H is an admissible domain if the following condi- tions are satisfied: (a) Every sequence h E H possesses a subsequence which belongs to D. (b) If h ED, and if g E H is equivalent to a subsequence of h, then g ED. Evidently, H itself is an admissible domain, and so are all sets HA with A E L$(E, P).
- 46. 26 Chapter 1. Limit Operators Proposition 1.3.5 The intersection of an at most countable set of admissible do- mains is an admissible domain again. Proof. Let D I, D2 , ... be admissible domains and D* := nkDk. It is evident that D* satisfies property (b) of an admissible domain. We claim that D* is subject to condition (a). Let h E H. Since D I is admissible, there is a subsequence hI of h which belongs to D I. Analogously, there is a subsequence h2 of hI which belongs to D2 , and so on. We form a new sequence h* by h*(k) := hk(k). This sequence belongs to D*. Indeed, the sequence h* is equivalent to a subsequence of the subsequence (h*(k+l), h*(k+2), ...) of hk. Hence, h* belongs to Dk for every k due to property (b) of the admissible domain Dk . 0 Let :Pi: denote the set of all bounded functions X which are defined on an admis- sible domain D(X) <;;:; 1t and take values in L(E), and which own the following property: If h E D(X) and if g E H is equivalent to a subsequence of h then X(h) = X(g). For example, all functions smboA defined by (1.36) belong to :Pi:. We provide :Pi: with operations as follows: Given X, Y E :Pi:, the sum X +Y (resp. the product XY) is the function which is defined on D(X) nD(Y) and takes the value X(h) +Y(h) (resp. X(h)Y(h)) at h E D(X) n D(Y). Further, if a E C, then aX is the function which is defined on D(X) and takes the value aX(h) at h E D(X). Finally, we set IIXII := sup{IIX(h)ll, hE D(X)}. Evidently, both operations satisfy the associativity and distributivity laws, and the addition is also commutative. The (additive) zero and (multiplicative) identity element are the functions 0: H -. L(E), h >---* 0 and I: H -. L(E), h>---* I with D(O) = D(I) = 1t respectively. But observe that, in general, X - X =I- 0 because of the different domains of definition of the functions X - X and O. Also, 11.11 satisfies the usual properties of a norm with the only exception that IIXII = 0 does not necessarily imply X = O. The announced equivalence relation on :Pi: which is aimed to solve these problems is defined by X rv Y if and only if XID(X)nD(Y) = YID(x)nD(Y). Proposition 1.3.6 The relation rv is an equivalence relation on :Pi: which is com- patible with the operations and with the norm. Proof. The reflexivity and symmetry of rv are evident. For the transitivity of rv, let X rv Y and Y rv Z, and let h E D(X) n D(Z). By property (a) of admissible domains, there is a subsequence 9 of h which lies in D(Y). Further, by property (b),
- 47. 1.3. Algebraization 27 9 belongs to every of the sets D(X), D(Y), D(Z). Now we have X(h) = X(g) = Y(g) because of X '" Y, and we have Z(h) = Z(g) = Y(g) because of Y '" Z. Hence, X(h) = Y(g) = Z(h) for all hE D(X) n D(Z). Let now Xl '" YI and X 2 '" Y2 . We will show that this implies Xl + X 2 '" YI +Y2 . The compatibility of", with the other operations follows analogously. What we have to check is the equality By definition, D(XI + X 2 ) = D(Xd n D(X2 ) and D(YI + Y2 ) = D(Yd n D(Y2 ). So, our claim is equivalent to Let hE D(Xd n D(X2 ) n D(YI ) n D(Y2 ). Then XI(h) = YI(h) since hE D(Xd n D(YI ) and Xl'" YI , X 2 (h) = Y2 (h) since hE D(X2 ) n D(Y2 ) and X 2 '" Y2 . Hence, XI(h) + X 2 (h) = YI(h) +Y2 (h). Finally, if W(X) ~ L(E) denotes the set of the values of the function X, then X '" Y implies that W(X) = W(Y). Indeed, if A E W(X), then there is an hE D(X) with X(h) = A, and we can find a subsequence 9 of h which is in both D(X) and D(Y). Hence, A = X(h) = X(g) = Y(g), i.e., A E W(Y). This shows that IIXII = IIYII whenever X '" Y. 0 Let :FE denote the set of all equivalence classes X~ of elements of J1 with respect to the equivalence relation "'. By the preceding proposition, it is correct to define operations on :FE by X~ + Y~ := (X + Y)~, X~Y~:= (XY)~, O'X~:= (O'X)~ and a norm on :FE by IIX~II :=IIXII· Proposition 1.3.7 :FE is a Banach algebra. Proof. The proof is straightforward. We will only check that IIX~ II = 0 implies X '" 0 and that the normed space :FE is complete. Let IIX~II = 0 and let X be a representative of the coset X~. Then X = 0 on D(X) and XID(X)nD(O) = XID(X) = OID(X) = OID(X)nD(O)' hence, X '" O. Together with the above remarks this shows that :FE is a normed algebra.
- 48. 28 Chapter 1. Limit Operators Let now (X;) be a Cauchy sequence in FE, choose representatives Xn E X;, and set D* := nnD(Xn). The Cauchy property of (X;) implies that, given c > 0 there is an no such that IIXnID*- XmID*11 :'::: c for all m, n ~ no. (1.38) In particular, (Xn(h)) is a Cauchy sequence in L(E) for every h E D*. Hence, (Xn(h)) is convergent, and we can define a function X on D* by X(h):= lim Xn(h) for every h E D*. n--->oo The function belongs to ~. Indeed, D(X) = D* is an admissible domain due to Proposition 1.3.5, and if g is equivalent to a subsequence of h E D*, then Xn(g) = Xn(h) for all n and, thus, X(g) = X(h). Our next goal is the boundedness of X. From (1.38) we conclude sup IIXn(h) - Xm(h)11 :'::: c for all m, n ~ no. hED* Letting m go to infinity yields sup IIXn(h) - X(h)11 :'::: c for every n ~ no hED* (1.39) and, consequently, IIXII :'::: IIXnID*11 + c :'::: IIXnl1 + c. So we have X E ~, and the convergence of X; to X~ in FE follows from IIX; - X~II = sup IIXn(h) - X(h) II hED* and (1.39). o In case E is a Hilbert space, one can define an involution on FE in an obvious way which makes FE to a C*-algebra. Definition 1.3.8 The symbol of the operator A E L$(E, P) is the coset smb A := (smboA)~ with smboA defined by (1.36). This definition is correct because riA is an admissible domain and smboA E :Pi; for all A E L$(E, P). Corollary 1.3.9 The mapping smb : L$(E, P) ---+ FE is a continuous algebra ho- momorphism. Proof Let us check, for example, that smb A+smbB = smb (A+B). By definition, smboA +smboB is the function, which is defined on riA nHB by h ~ Ah + Bh , and smbo(A + B) is the function defined on HA+B by h ~ (A + Bk Both functions coincide on HA n HB n HA+B. 0 Corollary 1.3.10 If A E L$(E, P) is P-Fredholm, then smbA is invertible in FE.
- 49. 1.4. Comments and references 1.4 Comments and references 29 The first appearance of limit operators is in Favard's paper [53], where they are used to verify the existence of almost-periodic solutions of ordinary differential equations with almost-periodic coefficients. Later, Muhamadiev [109, 110] applied limit operators to the question of solvability of elliptic partial differential equations in IRn . The limit operators method has been developed further in the papers [93, 95, 94, 118, 126, 137] for the study of the Fredholm property of wide classes of pseudodifferential operators and convolution operators on IRn and zn. Note also the paper [27], where the limit operators method has been applied to the computation of the essential spectrum of singular integral operators on Carleson curves acting in general weighted L2 -spaces. See also the monograph [72] for lots of applications in numerical analysis. We will give impressions of these results in Chapters 3, 4 and 6. Observe that in all of these papers, the method of limit operators is applied to a concrete class of operators acting on a concrete Banach space. Generalized notions of compactness have been introduced in [31, 52, 149], for example. Theorem 1.1.9 and its proof are (with minor modifications) taken from Kozak and Simonenko [87]. The original formulation of their result says that the inverse of an operator of local type is of local type again. It has been already employed in [149] to study the stability of approximation methods on spaces with supremum norm. There are other algebras besides L(E, P) which can be associated with an ap- proximate projection P in a natural way. A good candidate would be OLT(E, P), the algebra of all operators A E L(E) with IlPnA - APnII ---> 0 as n ---> 00. One easily checks that OLT(E, P) is a closed subalgebra of L(E) and that K(E, P) ~ OLT(E, P) ~ L(E, P). Moreover, it is evident that the algebra OLT(E, P) is in- verse closed in L(E). So, working with operators in OLT(E, P) seems to be much easier than working with operators from L(E, P). We give L(E, P) preference over OLT(E, P) for at least two reasons: L(E, P) is not only the largest algebra for which the P-compact operators form an ideal; this algebra is also (in contrast to OLT(E, P)) invariant with respect to the substitution of P by an equivalent approximate projection. Thus, when dealing with operators in L(E, P), we can switch over between equivalent approximate projections and choose that one which fits to our actual purposes.
- 50. Chapter 2 Fredholmness of Band-dominated Operators In this chapter we introduce a class of operators for which the converse of Corollary 1.3.10 can be proved: if the symbol of the operator is invertible, then the operator is P-Fredholm. The class under consideration consists of band and band-dominated operators which act on [P-spaces over ZN. In the forthcoming chapters we will point out that this class is large enough to include, e.g., discretizations of convolution operators and of pseudodifferential operators. 2.1 Band-dominated operators We start with fixing the Banach spaces E, the approximate identities P, and the group actions V, which will be used throughout this chapter. Then we continue with introducing the basic objects of this chapter: band and band-dominated operators with operator-valued entries. The main result of the section is Theorem 2.1.6, which provides us with different characterizations of band-dominated operators. 2.1.1 FUnction spaces on ZN For N a positive integer, let ZN denote the set of all N-tuples x = (Xl, ... , XN) of integers, provided with the norm Ixi = Ixloo := max{lxII, ... , IXNI}, and let X stand for a fixed complex Banach space. For p;:: 1, let [p(ZN,X) and [oo(ZN,X), respectively, stand for the Banach space of all functions I on ZN with values in X such that IIIII~:= L 111(x)ll~ < 00 and 1111100:= sup 111(x)llx < 00. xEZ N xEZ N Let further cO(ZN, X) refer to the closed subspace of [oo(ZN, X) which consists of all functions I with lim 111(x)llx = o. x--->oo In case X = C, we will simply write [p(ZN) and eo(ZN) in place of [p(ZN, X) and eo(ZN,X).
- 51. and Chapter 2. Band-dominated Operators For 1 :::; p < 00, the dual space of 1P(ZN, X) can be identified with 1q(ZN, X*) where l/p +1/q = 1, and the dual of co(ZN, X) is isomorphic to 11(ZN, X*). If, in particular, X is a reflexive Banach space, then the spaces 1P(ZN, X) are reflexive for all 1 < p < 00. If X = H is a Hilbert space, then 12(ZN, H) is a Hilbert space with respect to the inner product (1, g):= L (1(x), g(X))H' xEZN In what follows, we agree upon using the notation Eoo to refer to one of the spaces 1P(ZN,X) with 1 :::; p:::; (X) or to CO(ZN,X), whereas the symbol E will be used if only the spaces 1P(ZN, X) with 1 < p < (X) or cO(ZN, X) are taken into consideration. Our main emphasis will be on the spaces E, whereas the spaces 11(ZN,X) and 100(ZN,X) will playa major role only in Section 2.5, which is devoted to operators in the Wiener algebra. Given m E ZN, let 8 m stand for the function on ZN which is I E L(X) at m and 0 at all other points. The corresponding operator on L(Eoo ) of multiplication by 8 m will be abbreviated to Sm' For n 2: 0, we define Pn as the sum 2:lml:::;n Sm and we set Qn := 1- Pn . The operators Pn and Qn are projections, and QnPm = PmQn = 0 whenever n > m. Hence, the family P := (Pn ) constitutes a uniform approximate identity on each of the spaces Eoo, and P is a perfect approximate identity for all spaces E. It is non- perfect but still symmetric for the space 11(ZN,X), whereas it is neither perfect nor symmetric for 100(ZN,X). To see the latter, let f E 100(Z)* be a functional of norm one which vanishes on co(Z). The existence of such functionals is guaranteed by the Hahn-Banach theorem. Then (P~f)(x) = f(Pnx) = 0 for all x E 100(Z) and all n, whence the non-symmetry of P. Moreover, the same functional also yields a compact operator which does not belong to L(loo(Z), P). Indeed, choose a non- zero constant function g E 100(Z) and consider the rank one operator K x := f(x)g. Then PmKQnx = f(Qnx)Pmg = f(x)Pmg is independent of n and non-zero if x is accordingly chosen. Finally, for k E ZN, let Vk refer to the shift operator (Vkf)(X) = f(x - k), x E ZN. (2.1) Clearly, Vk E L(Eoo) and IlVkIIL(Eoo) = 1, and (1.32) as well as (1.33) are satisfied. Thus, the family V := {Vk : k E ZN} constitutes a group action on Eoo in the sense of Section 1.2.1. 2.1.2 Matrix representation It is often convenient to think of the operators in L(Eoo) as matrices with entries in L(X). For n E ZN, consider the restriction and extension operators Rn : ImSn -+ X, (... ,0, Xn, 0, ...) f--? Xn En: X -+ ImSn, Xn f--? ( ••• ,0, Xn, 0, ...),
- 52. 2.1. Band-dominated operators 33 with the X n standing at the nth place in the sequence. With every operator A E L(EOO ), we associate the matrix (Aij)i,jEZN where Aij := RiSiASjEj. Then, if U := (Uj)jEZN E Eoo is a sequence with finite support, the vector Au =: v = (Vi )iEZN is given by Vi = RiSiAu = RSiA L SjEjUj = L AijUj, j j thus, A acts as a usual matrix. The more interesting question is: Given two op- erators A, B E L(EOO) with the same matrix representation, is then necessarily A = B? Of course, the answer is no in general: The matrix representation of the operator K E L(lOO) considered at the end of Section 2.1.1 is the zero matrix, but K -I- o. Proposition 2.1.1 Both the operators in L(E) and the operators in L(EOO , P) are uniquely determined by their matrix representation. Proof. First observe that, for every operator A E L(EOO ) and for each n, the matrix representation (Aij ) of A determines the operators PnAPn completely. This follows from Pn = Llml:Sn Sm and PnAPn = L SiASj = L EiAj RjSj. lil,ljl:Sn lil,ljl:Sn Now, if p < 00, then the projections Pn converge strongly to the identity operator. Hence, PnAPn ~ A strongly for every A E L(E), showing that A is uniquely determined by its matrix representation. If p = 00, the Pn converge P-strongly to I. Thus, PnAPn ~ A P-strongly for every operator A E L(Eoo, P), what again yields the assertion. 0 2.1.3 Operators of multiplication Let loo ('J:F, L(X)) stand for the Banach algebra of all functions a on ZN with values in L(X) and Iialloo := sup Ila(x)IIL(x) < 00, xEZN and denote by CO(ZN, L(X)) the closed ideal of loo(ZN, L(X)) which consists of all functions a with lim Ila(x)IIL(X) = O. x-'oo We abbreviate loo(ZN,L(C)) to loo(ZN), and we will often identify a function a E loo(ZN) with the function x 1--+ a(x)I E L(X) which belongs to loo(ZN,L(X)). Every function a E loo(ZN, L(X)) gives rise to an operator on Eoo via (af)(x) = a(x)f(x), x E ZN. We call this operator the operator of multiplication by a and denote it by aI. Evidently, aI E L(Eoo) and IlaIIIL(E=) = lIalloo .
- 53. 34 Chapter 2. Band-dominated Operators Proposition 2.1.2 For a E lOO(ZN, L(X)), the operator aI belongs to L(EOO , P). This operator belongs to K(EOO , P) if and only if a E cO(ZN, L(X)). Proof. Let K E K(EOO, P). Since aI commutes with every operator Qn, we have IlaKQnll:::; IlaIllllKQnl1 ~ 0 and IIKaQnl1 = IIKQnaIl1 :::; IIKQnllllaIl1 ~ 0 as n ~ 00. Similarly, IIQnKaIl1 ~ 0 and IIQnaKl1 ~ 0, whence aI E L(EOO, P). For the second assertion observe that aI E K(EOO , P) if and only if, given c > 0, there is an no E N such that IlaQnl1 :::; E: for all n 2: no or, equivalently, such that Ila(x)11 :::; E: for all x with Ixl > no. The latter is equivalent to a E cO(ZN, L(X)). 0 We proceed with two equivalent characterizations of multiplication operators. Proposition 2.1.3 An operator A E L(EOO) is an operator of multiplication by a function in lOO(ZN, L(X)) if and only if ASm = SmA for allm E ZN. (2.2) Proof. It is evident that (2.2) holds if A is a multiplication operator. For the reverse implication, let A be subject to (2.2), and let En and Rn be defined as in Section 2.1.2. By MA we denote the operator of multiplication by the function a: ZN ~ L(X), m ~ RmSmASmEm. Evidently, a E lOO(ZN,L(X)), and for all m E ZN and all U = (Um)mEZN E Eoo, we have SmMAU = Ema(m)um = EmRmSmASmEmum = SmASmu = SmAu due to (2.2). This necessarily implies that MAu = Au for all U E Eoo, i.e., A is a multiplication operator. 0 Thus, an operator is an operator of multiplication if and only if its matrix repre- sentation is a diagonal matrix. For another criterion, let t = (h, ... , tN) E ]RN, consider the function et which is defined at x = (Xl, ... , XN) E ]RN by (2.3) and denote the restriction of a function a on ]RN onto ZN bya. Theorem 2.1.4 (a) An operator A E L(E) is the operator of multiplication by a function in lOO(ZN, L(X)) if and only if (2.4) (b) An operator A E L(E oo , P) is the operator of multiplication by a function in lOO(ZN, L(X)) if and only if (2.4) holds.
- 54. 2.1. Band-dominated operators 35 Proof. Clearly, every multiplication operator satisfies (2.4). For the reverse direc- tion, we claim that if an operator A E L(Eoo ) is subject to condition (2.4), then AbPn = bAPn for every bounded continuous function b on IR.N . (2.5) Once this is verified, and if A E L(Eoo, P), we take the P-strong limit of both sides of (2.5) which yields AbI = bA for all b and, hence, ASm = SmA for all m. Then assertion (b) follows from Proposition 2.1.3. In case A E L(E), we can take the usual strong limit in place of the P-strong one which yields the conclusion without further restriction on A. To verify (2.5), let £ denote the closure with respect to the supremum norm of the linear hull of all functions et, and let M n := [-n, n]N. Then, due to (2.4), aA = AM for all a E £. (2.6) The WeierstraB' approximation theorem implies that the restriction of £ onto M n is all of C(Mn ). Hence, if b is a bounded continuous function on IR.N then, for every fixed n, there is a function a E £ such that alMn = bl Mn . Consequently, AbPnf = Aapnf = aAPnf for all f E E oo (2.7) due to (2.6). Further, the function a can be chosen such that it takes at x E ZNMn an arbitrarily prescribed value. Since the left-hand side of (2.7) is independent of a, this shows that the function APnf vanishes outside ZN n Mn. Hence, AbPnf = aAPnf = bAPnf for all f E E oo which proves our claim (2.5). 0 There are operators on L(loo(Z)) which satisfy (2.4), but which fail to be multi- plication operators (see [91], Remark 2.1.9). 2.1.4 Band and band-dominated operators The basic objects of our interest are band and band-dominated operators. They are constituted by operators of multiplication and by the operators of shift introduced in (2.1). Definition 2.1.5 A band operator is a finite sum of the form La aaVa where a E ZN and aa E loo(ZN,L(X)). A band-dominated operator is the uniform limit of a sequence of band operators. To justify this notation note that, in case X = C. and N = 1 and with respect to the standard basis of E oo , band operators are given by matrices with finite band width. Observe also that the class of band operators is independent of the concrete space Eoo, whereas the class of band-dominated operators depends heavily on E oo . We denote this class by AEoo.
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- 60. This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook. Title: Rakkautta ja politiikkaa: Huvinäytelmä 1:ssä näytöksessä Author: Armas E. Turunen Release date: September 17, 2016 [eBook #53068] Language: Finnish Credits: Produced by Tapio Riikonen *** START OF THE PROJECT GUTENBERG EBOOK RAKKAUTTA JA POLITIIKKAA: HUVINÄYTELMÄ 1:SSÄ NÄYTÖKSESSÄ ***
- 61. Produced by Tapio Riikonen RAKKAUTTA JA POLITIIKKAA Huvinäytelmä 1:ssä näytöksessä Kirj. A. T. [Armas E. Turunen] Seuranäytelmiä N:o 30.
- 62. Hämeenlinnassa, Boman & Karlssonin kustannuksella, 1906. Hämeenlinnan Uudessa Kirjapainossa.
- 63. HENKILÖT: Ketonen, varakas suutarimestari, leski. Hilja, hänen tyttärensä. Peltola, puuseppämestari, vanhapoika. Einar Salmela, nuori konttoristi. Heta, Ketosen taloudenhoitajatar. Ananias, Ketosen sälli. Posteljooni. Tapahtuu eräässä pikkukaupungissa.
- 64. NÄYTTÄMÖ: Varakkaasti kalustettu huone Ketosella. Ovet perällä ja oikealla. Vasemmalla ikkuna ja sen vieressä sohva. Sohvan edessä pieni pöytä. 1 KOHTAUS. Ketonen. Ketonen (Yksin; istuu sohvalla aamutakki päällään ja myssy päässä, poltellen pitkää piippua ja katsellen pöydällä olevaa valokuvaa). Niin aivan tuollainen olit eläessäsi, Briittamuoriseni… tuollainen vähän kaareva nenä… hieman ryppyjä suun ympärillä… ne ovat ankaruuden merkkejä… Olithan joskus minulle vähän ankara, mutta en minä sentään mikään tohvelisankari ollut, vaikka pahat ihmiset niin sanoivat… sentään noudatin aina tahtoasi. Aina säilytän sinut rakkaassa muistossa… aina muistoasi kunnioitan, en voi Hiljaltanikaan kieltää mitään, kun hän on aivan samannäköinen kuin sinä olit nuorena… Sinä olet ollut poissa jo lähes 8 vuotta… Hilja on kasvanut sill'aikaa suureksi tytöksi… kohta hän menee naimisiinkin, miehen olen hänelle valinnut… niinhän se on tämän maailman meno… (Vetelee savuja), Hoh, hoi… eiköhän Hetalla jo olisi kahvi
- 65. valmiina? (Huutaa) Tuoppas, Heta, kahvini! (Heta ulkopuolella: Antaahan tuon nyt vähän selvitä!) Briittavainajalla oli aina kahvi valmiina heti ylös noustuani… Niinhän se on, ei saa ajoissa kahviaankaan, kun pitää laiskoja piikoja. (Katsoo kelloaan) Kah, johan kello on 9. Kohta tulee postikin ja saan taas lukea rakkaan Suomettareni. Se on epäilemättä paras lehti, vaikka sitä niin paljon parjataan. 2 KOHTAUS. Ketonen. Heta. Heta (Tulee perältä tuoden kahvitarjottimen). Jo tuo lienee selvää ja ei se niin vaarallista ole, jos vähän olisi poroakin… siitähän se rahakin maksetaan. Ketonen. Vieläköhän Hilja maannee, kun ei tule kahville…? Heta (Ivallisesti). No kuinkas muuten, kellohan on vasta 9. Väsyttäähän se tietysti, kun yökaudet juostaan kaikissa maailman seuroissa ja harjoituksissa ja tiaattereissa. Ei maar minun nuoruuteni aikana olisi sellaista sallittu ja ristin siitä tämänkin talon neidistä saa vielä ajanpitkään — sen minä sanon. Ketonen. Mutta eihän se niin vaarallista ole kuri… Heta. Vai ei ole vaarallista! vähäkö häntä alussa koetitte kieltää, mutta ettepäs saaneet tottelemaan. Ketonen. Enhän minä ainoalta lapseltani voi kieltää niin vähäpätöistä huvia… hän muistuttaa niin paljon äitivainajastaan ja…
- 66. Heta. Kyllä minä opettaisin tytön, jos minulla olisi valta. Kun hänelle antaa perään vähäisissä asioissa, niin sitten ei saa suuremmissakaan asioissa tottelemaan ja noudattamaan tahtoaan. Ketonen (Hieman suuttuen). No en tarvitse tässä nyt sinun neuvojasi… parasta on, että pidät huolen omista askareistasi. Heta (Kiivastuen). Enkö sitten ole tehnyt tehtäviäni!… Kyllähän minä tästä pois pääsen, jos ei työni kelpaa… Olen minä ollut paremmissakin paikoissa, eikä ole liikoja moitittu. (Nyyhkyttää). Ketonen. No, no, elähän nyt kiivastu, Heta! Enhän minä mitään pahaa tarkoittanut… Mene ja aja Hilja kahville. Kyllä on jo aika nousta. (Heta menee perälle. Ketonen juo kahvin). 3 KOHTAUS. Ketonen. Ketonen (Yksin). Saattaa tuo Heta sentään olla oikeassa… Olen kenties antanut Hiljalle liiaksi vapautta… Voi olla vaarallista, että on liiaksi myöntyväinen… Ei, en voi uskoa, ettei hän tottelisi isäänsä. Hän on kelpo tyttö ja Peltola veliseni tulee saamaan hänestä kelpo puolison… Onhan Peltola jo tosin vanhanpojan ijässä ja vähän änkyttää, mutta eihän ne niin suuria vikoja ole, ettei Hilja hänestä huolisi, kun minä vielä tahdon. (Posteljooni tulee perältä).
- 67. 4 KOHTAUS. Ketonen. Posteljooni. Posteljooni. Hyvää päivää mestari Ketonen! Tässä olisi "Suometar" (Antaa lehden) ja… ja… (Etsii laukusta) pari kirjettäkin on täällä. "Herra mestari K. Ketonen" ja "Neiti Hilja Ketonen". (Antaa kirjeet). Ketonen. Kiitoksia vaan… Posteljooni. Ai tosiaankin! Eikös Ananias Hytönen ole teillä sällinä? Ketonen. On kyllä. Onkos hänellekin jotain? Posteljooni. On tässä "Työmies", lienee sen tilannut ja nyt se tulee ensikertaa… Ketonen. "Työmies"! Kuka hänelle on antanut luvan sitä tilata? Posteljooni. Saahan sen jokainen omalla luvallaan… Ketonen (Ankarasti). Mutta minä sanon, että hän ei saa sitä tilata ja te ette saa sitä tänne tuoda… Posteljooni. Jaa, minun virkaani kuuluu kantaa jokaisen kotiin, mitä konttorista annetaan ja sitä mestari Ketonen ei voi kieltää… Hyvästi! (Menee). Ketonen. Mutta minä sanon, että ei! 5 KOHTAUS.
- 68. Ketonen. Sitten Hilja. Ketonen (Panee kirjeet pöydälle). "Työmies!" En voi kärsiä silmieni edessä tuota roskaista kiihoittajalehteä… se ei saa minun talooni tulla… ei! Ja Ananias, paras työmieheni, ainoa johon voin poikkeuksetta luottaa on tilannut tuon… se on kiihoittajain työtä… mutta kyllä minä hänestä perkaan sosialistiset ajatukset niin totta kuin nimeni on Ketonen. (Hilja tulee perältä) Kah, siinähän sinä jo oletkin tyttöseni. Hilja (Iloisena). Hyvää huomenta, isäkulta. (Syleilee häntä) Tiedätkös, minä tulen esiintymään Työväenyhdistyksen iltamassa näyttämöllä. Sinun pitää välttämättömästi tulla sitä katsomaan. Ketonen (Kummastuen). Mi-mitä! Minäkö rehellinen mies työväenyhdistyksen iltamaan? Se ei tule koskaan tapahtumaan. Sillä minä en voi kärsiä sosialistien vehkeitä. Sielläkös sinä olet käynytkin, tyttöseni? Hilja. Niin! Onko siinä sitte mitään pahaa? Sinunkin pitäisi kerran käydä siellä… saisit heistä toisen… Ketonen. Voi tyttöparkani, vai ovat he jo villinneet sinutkin. Hilja. Ei, isäni, ei siellä ketään villitä. Jospa kerran kuulisit Einarin puhuvan… Ketonen. Einarin?? Hilja. Niin, tarkoitan Einar Salmelaa… Ketonen. Hän juuri on pääkiihoittaja… itse piru ihmisen haamussa.
- 69. Hilja (Kiivaasti). Isä! Kuinka te voitte hänestä sanoa sellaista, hän on paras miehistä… Ketonen (Vakavasti). Kuuleppas tyttöseni! Sinun ei sovi millään tavalla olla tekemisissä sosialistien kanssa… se on vakava sanani… ja sitten älä puhu mitään tuosta Salmelasta, hän on vaarallinen henkilö. Hilja. Kyllä sinä vielä muutat mielesi, kunhan opit lähemmin tuntemaan Einarin. Ketonen. Mutta mitäs Peltola… (Erikseen) ah, ei vielä! (Ääneen) Mutta minä en tahdo sinun olevan missään tekemisissä hänen kanssaan. Hilja (Erikseen) Mitähän isä Peltolalla tarkoitti? (Ääneen) Pöydällä näkyy olevan kirjeitä. (Katsoo päällekirj.) Ketonen. Ai tosiaankin! En ollut muistaakaan… Anna minun kirjeeni. (Hilja antaa kirjeen). Hilja. Tämä on minulle, mutta en tunne käsialaa. Keltähän tämä mahtanee olla? Ketonen (Erikseen). Se on Peltolalta! (Ääneen) En minäkään ole ennen tämän kirjoittajalta kirjettä saanut… Hilja (Iloisesti). Minäpä tiedän keltä se on!… Se on… se on… Ketonen. Keneltä?
- 70. Hilja. Se on… enpäs sanokaan, lukekaa niin näette! Ketonen. Kohtahan siitä selvä saadaan… Menen huoneeseeni saadakseni sen rauhassa lukea… Juohan sinäkin kahvisi ja lue kirjeesi. (Riisuu aamutakin ja myssyn päältään ja panee ne tuolille; pukee tavallisen takin päälleen ja aikoo mennä). Ai, "Työmies" minun pitää… Hilja. Mitä! Työmies! Oletko tilannut "Työmiehen?" Ketonen. Sen on tilannut Ananias, mutta minä pistän sen uuniin. Hilja (Kiivaasti) Anna lehti tänne, isä. Sehän ei ole sinun! Ketonen (Pelästyen). Tuossa on, tuossa on, vaan… (Menee). 6 KOHTAUS. Hilja. Hilja. (Yksin, avaa kirjeensä). Keneltäs tämä nyt on? Katsotaanpas lopusta. "Henrik Hilarius Peltola, puuseppämestari". (Nauraa) Tämäpä vasta juhlalliselta kuuluu! Mitähän tuolla vanhanpojan hupelolla lienee minulle asiaa? (Lukee) "Lähestyn teitä tällä kirjeellä ja pyydän saada tunnustaa palavan rakkauteni teitä kohtaan". No jopas nyt jotakin!… "Enhän ole vielä liian vanhakaan, ainoastaan 48 vuotta. Isänne kanssa olen asiasta keskustellut ja hän on luvannut teidät minulle. Ajatelkaa tarkoin asiaa. Tulen tänään käymään teillä".
- 71. Teitä aina muistava Henrik Hilarius Peltola, puuseppämestari. (Purskahtaa kovasti nauramaan). Kaikkea sitä pistääkin vanhan pojan päähän ha, ha, ha! Kylläpä on Peltola parka erehtynyt… Ikää on hänellä vaan 48 vuotta! Jopa olisi ennen pitänyt pitää huolta kosimisesta… Mitähän Einar sanoo, kun näkee tämän kirjeen?… Hän varmaankin nauraa katketakseen… No enpä olisi uskonut!… (Nauraa). 7 KOHTAUS. Hilja. Heta. Heta (Tulee). No onpas nyt imeläsäkki leuan alla. Ei ole edes malttanut juoda kahviaankaan. (Hilja nauraa) Mikäs se nyt noin tirskuttaa? Hilja. Ihan kyyneleet tulevat silmiini… Et uskoisi Heta, hih-hih-hih… Et uskoisi! Heta. Kyllä minäkin jotain uskon! Hilja. Katsoppas tätä! (Antaa hänelle kirjeen). Heta. En minä hänestä ymmärrä tuon taivaallista! En osaa lukea kirjoitusta. (Antaa kirjeen takaisin). Hilja. Se on… (Nauraa) kosimakirje! Heta. Eihän sille pidä noin nauraa! Pitää ajatella asiaa tarkemmin.
- 72. Hilja. En saata olla nauramatta. Kosija on, näes, puuseppämestari Henrik Hilarius Peltola. (Nauraa). Heta. Henrik Hilarius Peltola. Kuka se on? Hilja (Nauraen). No etkö nyt häntä tunne…? Se änkyttävä mestari, joka usein käy meillä ja on minulle aina niin tavattoman kohtelias. Heta (Kummastuen). Hänkö! voi taivahan talikynttilät… Kyllähän minä änkyttävän mestarin tunnen… Ja hänkö nyt kosii neitiä? En uskoisi korviani… no, mitäs neiti nyt arvelee asiasta? Hilja. Aionpa vaan antaa hänelle pitkät… pitkät, pitkät rukkaset… Enhän minä ukosta huoli — en kohtakaan. Heta. Neidillä lienee toinen sulhanen? Hilja (Veikeästi). On! Heta. Kukahan tuo mahtanee olla? Hilja. Einar Salmela. Heta. Se anttikristuksen apostoli! Kuinka neitikin voi semmoisen miehen kanssa olla edes tuttavakaan? Hilja. Tulisitpas Heta, sinäkin kerran kuuntelemaan hänen puheitaan… Heta (Hämmästyneenä). Herra varjelkoon minua sinne menemästä! Ne ovat niitä maailman lopun enteitä ne "solisti"
- 73. kokoukset… Minun kotipitäjässäni kuuluisa saarna-Heikki varoitti sellaisista viettelijöistä… ne joutuvat armotta kadotukseen. Hilja (Nauraen). Heta parka, sinulla on aivan väärät luulot koko asiasta… tule kerran työväenyhdistykseen, jotta pääset selville… Heta (Menee kiireesti, vieden tarjottimen; itsekseen). Pakene kiusaajaa. 8 KOHTAUS. Hilja. Ketonen. Hilja. Eihän se mikään ihme ole, jos Heta ei työväenliikettä ja sen tarkotuksia ymmärrä, niitähän on paljon… paljon semmoisia ihmisiä… Ketonen (Pauhaten). Mestari Ketosen tyttö tuolle villitsijälle! viitsii tulla sellaista esittämäänkään! Hilja (Tietämättömyyttä teeskennellen): Mitä sinä nyt puhut?… Villitsijä?… Ketonen (Vihaisena). Olen saanut kirjeen Salmelalta, tuolta suurelta roistolta ja villitsijältä… hän pyytää sinua vaimokseen… sanoo sinun luvanneesi mennä hänelle. Hilja. Se on totta… Olen luvannut… Ketonen. Sinäkö olet luvannut? Etkö ole vielä lukenut kirjettäsi? Hilja. Kyllä jo luin…
- 74. Ketonen. Ja keneltä se oli? Hilja (Nauraen). Henrik Hilarius Peltolalta. Ketonen (Hyvillään). No siinä sitä nyt ollaan… hänelle sinun täytyy mennä, sehän on jotain, kun pääsee arvossa pidetyn mestarin rouvaksi… Hilja. Mutta minä en voi… rakastan Einaria ja… Ketonen. Hän on sosialisti… ja pitäisihän sinun sen verran ymmärtää… Hilja. Eikö teillä muuta syytä ole häntä vastaan? Ketonen. No, saattaahan hän muuten olla mies paikallaan mutta hän on sosialisti ja sinun ei mitenkään sovi… Hilja (Veikeästi). Minäpä olen itsekin sosialisti ja sitäpaitsi ei rakkauteen saa sekoittaa politiikkaa! Ketonen (Epätoivoisena). Niinkö pitkälle on tultu! He ovat villinneet ainoan lapsenikin… Heta oli oikeassa sanoessaan, että ei pitäisi lapselleen antaa niin suurta vapautta… Mutta ei vaikeroiminen enää auta… täytyy toimia pontevasti… Hilja. Mutta Peltolasta en huoli, en, en, vaikka mikä olkoon! Ketonen (Ankarana). Sinun täytyy! Minä olen se mies… Hilja. Isä! Mitähän äiti sanoisi jos olisi elossa. Ketonen (Hämmästyen). Taisin vähän kiivastua… mutta asioiden täytyy muuttua… menen heti Peltolan luo. (Erikseen mennessään)
- 75. Kun minä annoinkin hänelle niin paljon vapautta! (Menee). 9 KOHTAUS. Hilja. Einar Salmela. Hilja. Kyllä tästä vielä hyvä tulee… heti, kun mainitsin äidistä, niin masentui isä… luulenpa tästä selvittävän ilman muita mutkia… isä pitäisi vaan saada ymmärtämään, että rakkauteen ei saa sekoittaa politiikkaa. (Ovelle koputetaan) No, Peltola taitaa jo tulla… (Taas koputus) Sisään! (Salmela tulee perältä) Kas! Einarihan se olikin! (Tervehtivät). Einar. Kenenkä luulit tulevan, kultanuppuseni? Hilja (Nauraen). Sulhasen vaan! Einar. Sulhanenhan tulikin… Hilja. Niin, mutta… Einar. Mitä mutta…? Hilja. Luulin tulevan toisen sulhasen… (Istuutuvat sohvalle). Einar. Onko sinulla tyttöveitikalla useampiakin sulhasia? Hilja. Onpa kyllä… ja vielä oikein arvossa pidetty mestari.
- 76. Einar. En voi arvata… kukahan tuo mahtaisi olla? Hilja. Et voi olla nauramatta, jos sanon… Katsoppas tätä! (Antaa hänelle kirjeen; Einar lukee sen). No mitäs poika tuumii? Einar. Mitä ihmettä! Peltola — vanhapoika on ruvennut kosimapuuhiin! (Nauraa). Hilja. Hupsujahan ne vanhat pojat ovat!… Mutta mitäs siitä sanot, kun isäni kivenkovaan vaatii minua menemään naimisiin hänen kanssaan… sinä, kun olet sosialisti, niin hän ei salli minun mennä sinulle… Einar. Senkötähden? Eikö ole muuta syytä? Hilja. Ei minun tietääkseni. Einar. Siis rakkautta ja politiikkaa! Hilja. Älä sentään ole pahoillasi poikaseni, kyllä minä muutan isäni mielen. Einar (Syleillen Hiljaa). Kyllä sinä olet aika veitikka pikku enkelini… annathan suukkosen?… (Katsoo häntä hellästi silmiin). Hilja. Kaikkia vielä! (Einar suutelee. Hilja näppää häntä poskelle) Kas tuossa rangaistus, senkin huimapää! (Käytävässä kuuluu askelia) Ai! joku tulee! Mene piiloon!… Ei, pue päällesi isän aamutakki… nyt sukkelaan. (Auttaa Einaria hänen pukiessaan päälleen aamutakkia ja panee myssyn hänen päähänsä) Kas noin! Nyt ikkunan luo… muuta
- 77. ääntäsi… se on kai joku sälli. (Einar asettuu ikkunan luo ja katselee ulos.) Nyt! 10 KOHTAUS. Edelliset. Ananias. Ananias (Tulee oikealta suutarin työpuvussa, hihat käärittynä ylös). Herra mestari! Täällä olisi mestari Peltola ja hän tahtoisi tavata mestaria… Einar (Seisoen selin Ananiakseen, muuttaen ääntään). Mitä hänellä on asiaa? Ananias. En minä kysynyt hänen asioitaan… Hilja. En tahdo, että hän nyt tulee tänne, sano, että isä ei ole kotona… Ananias. Minä en koskaan valehtele… ja onhan mestari kotona… menen sanomaan. (Menee). Einar (Yrittää mennä Ananiaan jälestä). Mutta enhän minä mikään mestari ole… ah hän ehti jo mennä. Hilja. Ja nyt tulee Peltola tänne! Parempi olisi ollut, jos olisit piiloittautunut, silloin olisin voinut sanoa, että en ota vastaan. 11 KOHTAUS.
- 78. Hilja, Einar ja Peltola. Peltola (Tulee oikealta. Einar kääntyy taas katselemaan ikkunasta). Hy-hy-hyvää päivää, neiti Ke-Ketonen! (Tervehtii kohteliaasti) Kuinkas vo-voitte? Hilja. Kiitos kysymästänne, oikein hyvin! Peltola. Se-sepä on ha-hauskaa! Te-terve ve-veli Ke-Ketonen, vielähän si-sinä olet a-aamupuvussasi. (Menee tervehtimään). Einar (Kääntyy iloisesti ympäri). Terve, terve! Peltola (Hämmästyneenä). Mi-mitä tä-tämä on? Sa-salmela! Hilja. Minä vaan pyysin hänen panemaan päälleen isäni aamutakin… hän on niin hauskan näköinen sen sisällä… Peltola. Siis pi-pieniä ku-kujeita. Mu-mutta le-leikki sikseen… Hilja. Pyydän anteeksi, että poistun. Käsken vaan Hetan laittamaan kahvia. Olkaa hyvä ja istukaa! (Peltola ja Salmela istuutuvat sohvaan, Hilja menee perälle). 12 KOHTAUS. Peltola. Salmela. Einar. Teillä kai olisi ollut asiaa mestarille?
- 79. Peltola. Ei e-erittäin me-mestarille. Ne-neidin ka-kanssa minun pi- pitäisi pu-puhua. O-olen, näes, aikonut me-mennä na-naimisiin. Einar. Soo! Isä-vainajalleni aina vakuutitte pysyvänne vanhanapoikana. Peltola. Eihän si-sille mitään ma-mahda, jos ra-rakastuu… Einar. Olette siis rakastunut? Peltola. Tä-tämän ta-talon, neiti on vo-voittanut sy-sy-dämmeni. Ki-kirjotin hänelle, ja nyt tu-tulin py-pyytämään va-vastausta. E-etkö sinä ru-rupeaisi puhe-puhemiehekseni? Einar. Enhän minä… enhän minä voi, kun… 13 KOHTAUS. Edelliset. Hilja (Tulee). Peltola (Einarille). A-autahan nyt vä-vähä. (Hiljalle) Jo-joko ne-neiti on lu-lukenut ki-kirjeeni? Hilja. Jo luin. Peltola. No, o-onko mi-minulla siis to-toivoa? Hilja. Anteeksi, minun täytyy kieltää, olen luvannut mennä toiselle. Peltola. To-toiselle! Ku-kuka o-on tu-tuo to-toinen? Hilja. Einar Salmela.
- 80. Peltola. Ta-tahdotko sinä E-Einar riistää mi-mi-nulta mo- morsiamen? Einar. Me olemme jo rakastaneet toisiamme kauan… Hilja. En voi enään ketään toista rakastaa… Peltola (Erikseen). Mi-mi-minä en siis saa häntä! (Ääneen) I-isä- vainajasi oli pa-ras ystäväni… en ta-tahdo pa-pahoitttaa hänen po- poikansa mi-mieltä… mi-minäkin sen ve-verran ymmärrän, e-että sinä olet so-sopivampi Hi-hiljalle. O-ottakaa mi-minun pu-puolestani toisenne ja o-olkaa o-onnelliset. Einar. Kiitos, setä Peltola. Te olette jalomielinen! (Pudistaa Peltolan kättä). Hilja. Kiitos minunkin puolestani! Mutta vielä pyytäisin teiltä yhden palveluksen… Peltola Mi-mikä se o-olisi? Hilja. Pyytäisin teitä puhumaan puolestamme isälleni… tehän tunnette Einarin… Peltola. Ky-kyllä! Pa-parempaa mi-miestä ei Hi-Hilja voi sa-saada. Teen vo-voitavani. Hilja (On katsonut ikkunasta). Isä tulee! Peltola. Jo-joko hän tulee, ky-kyllä minä pu-puhun… Einar. Ja minä olen täällä!
- 81. (Riisuu kiireesti aamutakin ja myssyn ja panee ne tuolille.) Hilja. Mene sinä minun huoneeseeni, Einar! (Peltolalle) Ymmärrättehän! (Työntää Einarin edellään ulos perältä). Peltola (Nauraa). Ha-hauskoja ve-ve-veitikoita, mu-mutta mi- minun tä-täytyy jä-jäädä vanhaksi pojaksi. (Hilja tulee). Hilja. Älkää sanoko isälle mitään! Hän ei saa tietää Einarin olevan täällä. Peltola. Ky-kyllä y-ymmärrän! 14 KOHTAUS. Hilja. Peltola. Ketonen. Ketonen (Tulee perältä). Kas täällähän sinä oletkin veliseni ja minä kun kävin sinua etsimässä kotoasi. Olikin hyvä, että tulit. Hilja, meneppäs puuhaamaan kahvia! Hilja. Käskin jo Hetan keittää, kyllä se kohta joutuu. Ketonen. No mitenkäs ne asiat hurisevat, mitä sinulle kuuluu, veliseni. Peltola. E-eihän tä-tässä mitään e-erinäisiä, tu-tulin vaan ta- tapaamaan ty-tytärtäsi, kosimapuuhissa o-olen.
- 82. Ketonen (Hyvillään). Ja tyttäreni suostuu?… Johan sen sanoin… Peltola. E-ei su-suostu. Ketonen. Mitä? Paneeko hän vastaan. Hilja. Emmehän me sovi toisillemme ja minä kun… Peltola. Me pä-pä-päätimmekin, e-ettemme me-menekään na- naimisiin. Hilja. En voi, sillä rakastan toista. Olen, kuten jo sanoin, luvannut mennä Salmelalle. Ketonen. Hiljaa, tyttö! En kärsi kuulla puhuttavan tuosta Salmelasta… Peltola. Mi mi-mitäs sinulla o-on oikeastaan si-sitä mi-miestä vastaan. Minä hu-huomasin, vaikka my-myöhään, e-että e-ei minun so-sovi mennä na-naimisiin nuoren ty-tytön kanssa… Ketonen. Mutta ei sittenkään hän mene Salmelalle… Peltola. Mu-mutta ky-kyllähän se mies ja-jaksaa vaimonsa e- elättää on hä-hänellä siksi hy-hyvät tu-tulot. Hänet o-olen tu- tuntenut ka-kauvan, hän on hy-hyvä mies… mitäs sinulla si-sitten o- on hä-häntä vastaan. Ketonen. Hän on sosialisti… hän villitsee ihmisiä koettaa kiihoittaa ihmisiä laiskureiksi… sanalla sanoen en kärsi sosialisteja… Peltola. Minäkin ku-kuulun ty-työväen yhdistykseen… Hilja. Minä myös!
- 83. Ketonen. Mitä? Oletteko tulleet hulluiksi… onko Salmela saanut kaikki villityksi?… 15 KOHTAUS. Edelliset. Heta. Salmela. Ananias. Heta (Juoksee hengästyneenä perältä). Auttakaa, auttakaa! varkaita! Ketonen. Varkaita! Missä? Peltola. Va-varkaita? Heta. Hiljan huoneessa on varkaita!… Ketonen (Huutaa oikeanp. ovesta). Ananias! Tule sukkelaan! Peltola (Hiljalle). Nyt hän jo-joutui ki-kiikkiin. (Ananias tulee oikealta). Ketonen. Tule kanssani Ananias, otetaan varas kiinni. (Menevät). Hilja. Kyllä nyt täytyy saada isän suostumus! (Ananias ja Ketonen kulettavat Einarin sisään). Ketonen (Hämillään). Mitäs teillä täällä on tekemistä? Aijotteko minunkin talossani ruveta kiihoittamaan ja villitsemään rauhallisia ihmisiä. Se on varma, että tästä talosta ette saa ketään villityksi,
- 84. joskin tytärtäni olette narranneet. Ole varoillasi, Ananias, tämän miehen vehkeitä vastaan. Ananias. En ole ollut hänen kanssaan muussa tekemisessä, kun vaan "Työmies" lehden tilasin hänen kauttaan… Hilja (Ottaa pöydältä lehden ja antaa sen Ananiaalle). Tässä on lehtesi! Ketonen (Ankarana). Mutta minä en salli… Hilja. Isä! (Ananias menee oikealle). Ketonen. Vai hänenkin olette narranneet… paras työmieheni!… Einar. En ole ketään narrannut, se on vaan totta, että olen työväenyhdistyksen johtavia henkilöitä ja olen siellä pitänyt puheita ja koettanut parhaan kykyni mukaan selittää lähimmäisilleni sosialismin suurta aatetta, vaan se ei ole mitään villitsemistä, sen voivat nämä läsnäolevat todistaa, sillä he ovat olleet kuulemassa. Ketonen. Siitä sen nyt kuulee, olet vietellyt heidät seuraasi. Mutta minun omaisuuttani ette tule jakamaan… olen minä vaan se mies… Einar. Omaisuutta jakamaan!… Ketonen. Sehän se teidän sosialistien päätarkoitus juuri on! Mutta ei se vaan niin käy… Einar. Se on turhaa puhetta, jota aatteemme vastustajat ovat levittäneet ja jota jotkut ymmärtämättömyydessään uskovat.
- 85. Peltola. Ei mi-minunkaan o-omaisuuttani ku-kukaan ruvennut jakamaan… Einar. Se on kerrassaan aivan väärä luulo, sillä meillä on paljon jalompi tarkoitus… Tarkoituksemme on kohottaa köyhälistö siitä alennuksen ja sorron tilasta, jossa se tähän asti on ollut. Toivon teidän nyt ymmärtävän… Mutta tarkoitukseni ei ollut tulla teille mitään esitelmiä pitämään… tahdoin pyytää teiltä tytärtänne vaimokseni… me rakastamme toisiamme… älkää kieltäkö! (Ottaa Hiljan kädestä kiinni). Ketonen. Minäkö, mestari Ketonen, antaisin tyttäreni, sosialistille… ei, siitä ei tule mitään… Peltola. E-e-eihän sinulla ole mi-mitään mu-muuta häntä va- vastaan ja eihän ra-rakkauteen pi-pidä se-sekoittaa po-politiikkaa. Su-suostu va-vaan heidän py-pyyntöönsä! Ketonen. Se on mahdotonta!… parasta on siis että poistutte, herra Salmela. Hilja. Siis et suostu, isä? Ketonen. En! Hilja (Kiivaasti). Silloin poistun minä myös. Lähdetään Einar, kyllä me tulemme maailmalla toimeen… Heta (Erikseen). Kyllä nämä ovat maailman lopun enteitä! (Menee perälle).
- 86. Hilja. Lähtekäämme; mutta sen minä tiedän, että tätä ei olisi tapahtunut, jos äiti olisi elossa. Peltola. Vo-voitko la-laskea ty-tyttäresi me-menemään. A-anna vaan su-suostumuksesi. Mi-minä ta-takaan, että Hilja saa kelpo mi- miehen… (Hilja ja Einar yrittävät lähteä.) Ketonen (Neuvotonna). Mitähän Briitta olisi tehnyt tässä asiassa? Hilja. Ei hän vaan näin olisi antanut tapahtua. Peltola. Ei Briitta o-olisi sa-sallinut sekoittaa ra-rakkautta po- politiikkaan. Hilja (Avaa oven). Siis meidän täytyy lähteä… Peltola. Ve-veli Ke-Ketonen! Ketonen (Heltyneenä). Ei, älkää menkö. Olenpa kummallisessa asemassa. Mitä tehdä? (Ajattelee hetken) Hmh… hmh… No niin, eihän sille mitään mahda! Ottakaa lapset toisenne ja olkaa onnelliset! Einar. Kiitos! (Sulkee Hiljan syliinsä) Nyt olet minun! Peltola. Py-pyydän saada to-toivottaa onnea! (Kättelee). Hilja. Kiitos, kiitos jalomielisyydestänne, setä Peltola. (Keskustelevat hiljaa keskenään.)
- 87. Ketonen (Erikseen). Luulenpa melkein, että Briittakin olisi suostunut, jos hän olisi elossa, eikä hän olisi, kukaties, antanut sekoittaa rakkauteen politiikkaa. (Ääneen) Mutta sinulle minä sanon, veli Peltola, että sinä et tuolla tavoin toimien saa eukkoa kuuna päivänä. Peltola. Minä pä-päätinkin py-pysyä loppuikänikin va-vanhana po- poikana. Mutta vä-vähät si-siitä! Hy-hyvilläni o-olen ku-kun a-asiat nä-näin kä-kääntyivät, si-sillä va-vaarallista o-on se sekottaa ra- rakkauteen po-politiikkaa! (Esirippu alas.) End of Project Gutenberg's Rakkautta ja politiikkaa, by Armas E. Turunen
- 88. *** END OF THE PROJECT GUTENBERG EBOOK RAKKAUTTA JA POLITIIKKAA: HUVINÄYTELMÄ 1:SSÄ NÄYTÖKSESSÄ *** Updated editions will replace the previous one—the old editions will be renamed. Creating the works from print editions not protected by U.S. copyright law means that no one owns a United States copyright in these works, so the Foundation (and you!) can copy and distribute it in the United States without permission and without paying copyright royalties. Special rules, set forth in the General Terms of Use part of this license, apply to copying and distributing Project Gutenberg™ electronic works to protect the PROJECT GUTENBERG™ concept and trademark. Project Gutenberg is a registered trademark, and may not be used if you charge for an eBook, except by following the terms of the trademark license, including paying royalties for use of the Project Gutenberg trademark. If you do not charge anything for copies of this eBook, complying with the trademark license is very easy. You may use this eBook for nearly any purpose such as creation of derivative works, reports, performances and research. Project Gutenberg eBooks may be modified and printed and given away—you may do practically ANYTHING in the United States with eBooks not protected by U.S. copyright law. Redistribution is subject to the trademark license, especially commercial redistribution. START: FULL LICENSE
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