![216 B. Yousefi, M. Habibi
The first example of a hypercyclic operator on a Hilbert space was con-
structed by Rolewicz in 1969 (see [12]). He showed that if B is the backward
shift on 2(N), then λB is hypercyclic if and only if |λ| > 1.
The holomorphic self maps of U are divided into classes of elliptic and non-
elliptic. The elliptic type is an automorphism and has a fixed point in U. It is
well known that this map is conjugate to a rotation z → λz for some complex
number λ with |λ| = 1. The maps of that are not elliptic are called of non-
elliptic type. The iterate of a non-elliptic map can be characterized by the
Denjoy-Wolff Iteration Theorem (see [14]).
A complex-valued function ψ on G is called a multiplier of H if ψH ⊂ H.
The operator of multiplication by ψ is denoted by Mψ and is given by f −→ ψf.
By the closed graph theorem Mψ is bounded. The collection of all multipliers
is denoted by M(H). Each multiplier is a bounded analytic function on G. In
fact ϕ G ≤ Mϕ .
If w is a multiplier of H and ϕ is a mapping from G into G such that
f ◦ ϕ ∈ H for all f ∈ H, then Cϕ (defined on H by Cϕf = f ◦ ϕ) and MwCϕ
are called composition and weighted composition operator respectively. We
define the iterates ϕn = ϕ ◦ ϕ ◦ . . . ◦ ϕ (n times). For some related topics, see
[1]-[17].
2. Main Results
A nice criterion, namely the Hypercyclicity Criterion, is used in the proof of
our main theorem. It was developed independently by Kitai ([10]), Gethner
and Shapiro ([6]). This criterion has been used to show that hypercyclic op-
erators arise within the classes of composition operators ([4]), weighted shifts
([13]), adjoints of multiplication operators ([5]), and adjoints of subnormal and
hyponormal operators ([3]).
The formulation of the Hypercyclicity Criterion in the following theorem
was given by J. Bes in his PhD thesis (see [1], also [2]).
Theorem 1. (The Hypercyclicity Criterion) Suppose X is a separable
Banach space and T = (T1, T2) is a pair of continuous linear mappings on
X. If there exist two dense subsets Y and Z in X and two strictly increasing
sequences {nj} and {kj} such that:
1. T
nj
1 T
kj
2 y → 0 for every y ∈ Y , and
2. There exist functions Sj : Z → X such that for every z ∈ Z, Sjz → 0, and
T
nj
1 T
kj
2 Sjz → z,
then T is a hypercyclic pair.](https://image.slidesharecdn.com/2bf6f431-06e8-4d53-87b0-8ddbfeef6d0a-150724062249-lva1-app6891/85/PaperNo6-YousefiHabibi-IJAM-2-320.jpg)
![218 B. Yousefi, M. Habibi
for all λ in E−1. Note that
n−1
i=0
(w1 ◦ ϕi+n ◦ ϕ−1
2n .w2 ◦ ϕi ◦ ϕ−1
2n )(λ) =
n
i=1
(w1 ◦ ϕ−1
i .w2 ◦ ϕ−1
n+i)(λ).
Thus LnSn is identity on the dense subset HE−1 of H. Note that if λ ∈
E−1, then Sn −→ 0 pointwise on HE−1 that is dense in H. Thus the pair
((Mw1 Cϕ)∗, (Mw2 Cϕ)∗) satisfies the Hypercyclicity Criterion.
In the case that linear functionals of point evaluations are not linearly in-
dependent, by the same way we can use a standard method as in Theorem 4.5
in [7] to complete the proof.
If f is a function, by ran f we mean the range of f.
Corollary 3. Suppose that h is a nonconstant multiplier of H such that
ran h intersects the unit circle. Then the adjoint of the multiplication operator
Mh satisfies the Hypercyclicity Criterion.
Proof. In Theorem 2, let ϕ be identity and w1 = w2 = h. Now the sets
Fm = {λ ∈ D : |h(λ)|m < 1} are nonempty and open sets in D and so clearly
have limit points in D for m = −1, 1. But Fm ⊂ Em where Em = {λ ∈ D :
∞
i=1 h(λ)2m
= 0} for m = −1, 1. Now we can apply the result of Theorem 2,
and so the proof is complete.
References
[1] J. Bes, Three problems on hypercyclic operators, PhD Thesis, Kent State
University (1998).
[2] J. Bes, and A. Peris, Hereditarily hypercyclic operators, Jour. Func. Anal.,
167, No. 1 (1999), 94-112.
[3] P.S. Bourdon, Orbits of hyponormal operators, Mich. Math. Jour., 44
(1997), 345-353.
[4] P.S. Bourdon and J.H. Shapiro, Cyclic phenomena for composition opera-
tors, Memoirs of the Amer. Math. Soc., 125, AMS, Prov. - RI, 1997.
[5] P.S. Bourdon and J.H. Shapiro, Hypercyclic operators that commute with
the Bergman backward shift, Trans. Amer. Math. Soc., 352, No. 11 (2000),
5293-5316.](https://image.slidesharecdn.com/2bf6f431-06e8-4d53-87b0-8ddbfeef6d0a-150724062249-lva1-app6891/85/PaperNo6-YousefiHabibi-IJAM-4-320.jpg)
![HYPERCYCLICITY CRITERION FOR A PAIR OF... 219
[6] R.M. Gethner and J.H. Shapiro, Universal vectors for operators on spaces
of holomorphic functions, Proc. Amer. Math. Soc., 100 (1987), 281-288.
[7] G. Godefroy, J.H. Shapiro, Operators with dense, invariant cyclic vector
manifolds, Jour. of Func. Anal., 98 (1991), 229-269.
[8] K. Goswin, G. Erdmann, Universal families and hypercyclic operators,
Bull. of the Amer. Math. Soc., 35 (1999), 345-381.
[9] D.A. Herrero, Limits of hypercyclic and supercyclic operators, Jour. of
Func. Anal., 99 (1991), 179-190.
[10] C. Kitai, Invariant closed sets for linear operators, Thesis, University of
Toronto, 1982.
[11] C. Read, The invariant subspace problem for a class of Banach spaces, 2:
Hypercyclic operators, Isr. J. Math., 63 (1988), 1-40.
[12] S. Rolewicz, On orbits of elements, Studia Math., 32 (1969), 17-22.
[13] H.N. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc., 347
(1995), 993-1004.
[14] J.H. Shapiro, Composition Operators and Classical Function Theory,
Springer-Verlag, New York, 1993.
[15] B. Yousefi and H. Rezaei, Hypercyclic property of weighted composition
operators, Proc. Amer. Math. Soc., 135, No. 10 (2007), 3263-3271.
[16] B. Yousefi and H. Rezaei, On the supercyclicity and hypercyclicity of the
operator algebra, Acta Mathematica Sinica, 24, No. 7 (2008), 1221-1232.
[17] B. Yousefi and R. Soltani, Hypercyclicity of the adjoint of weighted com-
position operators, Proc. Indian Acad. Sci., 119, No. 3 (2009), 513-519.](https://image.slidesharecdn.com/2bf6f431-06e8-4d53-87b0-8ddbfeef6d0a-150724062249-lva1-app6891/85/PaperNo6-YousefiHabibi-IJAM-5-320.jpg)
PaperNo6-YousefiHabibi-IJAM
- 1. International Journal of Applied Mathematics ————————————————————– Volume 24 No. 2 2011, 215-219 HYPERCYCLICITY CRITERION FOR A PAIR OF WEIGHTED COMPOSITION OPERATORS B. Yousefi1, M. Habibi2 § 1Department of Mathematics Payame-Noor University Shahrake Golestan, P.O. Box 71955-1368, Shiraz, IRAN e-mail: bahmann@spnu.ac.ir 2Department of Mathematics Islamic Azad University, Branch of Dehdasht P.O. Box 7571763111, Dehdasht, IRAN e-mail: habibi.m@iaudehdasht.ac.ir Abstract: In this paper we give some sufficient conditions for a pair of the adjoint weighted composition operators acting on some function spaces satisfying the Hypercyclicity Criterion. AMS Subject Classification: 47B37, 47B33 Key Words: hypercyclic vector, hypercyclicity criterion, weighted composi- tion operator, multiplication operator 1. Introduction Let H be a Hilbert space of functions analytic on a plane domain G such that for each λ in G the linear functional of evaluation at λ given by f −→ f(λ) is a bounded linear functional on H. By the Riesz representation theorem, there is a vector Kλ in H such that f(λ) =< f, Kλ >. We call Kλ the reproducing kernel at λ. Let T1, T2 be commutative bounded linear operators on H and T = (T1, T2). Put F = {T1 m T2 n : m, n ≥ 0}. For x ∈ H, the orbit of x under T is the set orb(T, x) = {Sx : S ∈ F}. The vector x is called hypercyclic for T if orb(T, x) is dense in H. Received: December 21, 2010 c 2011 Academic Publications §Correspondence author
- 2. 216 B. Yousefi, M. Habibi The first example of a hypercyclic operator on a Hilbert space was con- structed by Rolewicz in 1969 (see [12]). He showed that if B is the backward shift on 2(N), then λB is hypercyclic if and only if |λ| > 1. The holomorphic self maps of U are divided into classes of elliptic and non- elliptic. The elliptic type is an automorphism and has a fixed point in U. It is well known that this map is conjugate to a rotation z → λz for some complex number λ with |λ| = 1. The maps of that are not elliptic are called of non- elliptic type. The iterate of a non-elliptic map can be characterized by the Denjoy-Wolff Iteration Theorem (see [14]). A complex-valued function ψ on G is called a multiplier of H if ψH ⊂ H. The operator of multiplication by ψ is denoted by Mψ and is given by f −→ ψf. By the closed graph theorem Mψ is bounded. The collection of all multipliers is denoted by M(H). Each multiplier is a bounded analytic function on G. In fact ϕ G ≤ Mϕ . If w is a multiplier of H and ϕ is a mapping from G into G such that f ◦ ϕ ∈ H for all f ∈ H, then Cϕ (defined on H by Cϕf = f ◦ ϕ) and MwCϕ are called composition and weighted composition operator respectively. We define the iterates ϕn = ϕ ◦ ϕ ◦ . . . ◦ ϕ (n times). For some related topics, see [1]-[17]. 2. Main Results A nice criterion, namely the Hypercyclicity Criterion, is used in the proof of our main theorem. It was developed independently by Kitai ([10]), Gethner and Shapiro ([6]). This criterion has been used to show that hypercyclic op- erators arise within the classes of composition operators ([4]), weighted shifts ([13]), adjoints of multiplication operators ([5]), and adjoints of subnormal and hyponormal operators ([3]). The formulation of the Hypercyclicity Criterion in the following theorem was given by J. Bes in his PhD thesis (see [1], also [2]). Theorem 1. (The Hypercyclicity Criterion) Suppose X is a separable Banach space and T = (T1, T2) is a pair of continuous linear mappings on X. If there exist two dense subsets Y and Z in X and two strictly increasing sequences {nj} and {kj} such that: 1. T nj 1 T kj 2 y → 0 for every y ∈ Y , and 2. There exist functions Sj : Z → X such that for every z ∈ Z, Sjz → 0, and T nj 1 T kj 2 Sjz → z, then T is a hypercyclic pair.
- 3. HYPERCYCLICITY CRITERION FOR A PAIR OF... 217 Throughout this section let H be a Hilbert space of analytic functions on the open unit disc D such that H contains constants and the functional of evaluation at λ is bounded for all λ in D. Also let wi : D → C be non-constant multipliers of H for i = 1, 2, and ϕ be an analytic univalent map from D onto D. By ϕ−1 n we mean the nth iterate of ϕ−1. Theorem 2. Suppose that the composition operator Cϕ is bounded on H and w1, w2 are such that w1.w2 ◦ ϕ = w2.w1 ◦ ϕ, and the sets E1 = {λ ∈ D : lim n n−1 i=0 w1 ◦ ϕi+n(λ).w2 ◦ ϕi(λ) = 0} and E−1 = {λ ∈ D : lim n ( n i=1 w1 ◦ ϕ−1 i (λ).(w2 ◦ ϕ−1 i+n(λ)) −1 = 0} have limit points in D. If for each λ ∈ Em the sequence {Kϕm 2i (λ)}i is bounded for m = −1, 1, then the pair ((Mw1 Cϕ)∗, (Mw2 Cϕ)∗) satisfies the Hypercyclicity Criterion. Proof. Put Ln = ((Mw1 Cϕ)∗)n((Mw2 Cϕ)∗)n. Then for all n ∈ N and all λ in D we get LnKλ = n−1 i=0 w1 ◦ ϕi+n(λ).w2 ◦ ϕi(λ) Kϕ2n (λ). Put HEm = span{Kλ : λ ∈ Em} for m = −1, 1. The set HEm is dense in H, because: if f ∈ H and < f, Kλ >= 0 for all λ in Em, then f(λ) = 0 for all λ in Em. So by using the hypothesis of the theorem, the zeros of f has limit point in D which implies that f ≡ 0 on D. Thus HEm is dense in H. Note that if λ ∈ E1, then lim n LnKλ = 0. Thus Ln −→ 0 pointwise on HE1 that is dense in H. Now to find the right inverse of Ln, first consider the special case where the collection of linear functionals of point evaluations {Kλ : λ ∈ E−1} is linearly independent. Note that in the following definition there is no possibility of dividing by zero. Define Sn : HE−1 −→ H by extending the definition SnKλ = ( n i=1 w1 ◦ ϕ−1 i (λ).w2 ◦ ϕ−1 n+i(λ))−1 )Kϕ−1 2n (λ), where ϕ−1 i is the ith iterate of ϕ−1 and n ∈ N. Now we have LnSnKλ = ( n i=1 (w1 ◦ ϕ−1 i .w2 ◦ ϕ−1 n+i)(λ))−1 . n−1 i=0 (w1 ◦ ϕi+n ◦ ϕ−1 2n .w2 ◦ ϕi ◦ ϕ−1 2n )(λ) Kϕ−1 2n ◦ϕ2n (λ)
- 4. 218 B. Yousefi, M. Habibi for all λ in E−1. Note that n−1 i=0 (w1 ◦ ϕi+n ◦ ϕ−1 2n .w2 ◦ ϕi ◦ ϕ−1 2n )(λ) = n i=1 (w1 ◦ ϕ−1 i .w2 ◦ ϕ−1 n+i)(λ). Thus LnSn is identity on the dense subset HE−1 of H. Note that if λ ∈ E−1, then Sn −→ 0 pointwise on HE−1 that is dense in H. Thus the pair ((Mw1 Cϕ)∗, (Mw2 Cϕ)∗) satisfies the Hypercyclicity Criterion. In the case that linear functionals of point evaluations are not linearly in- dependent, by the same way we can use a standard method as in Theorem 4.5 in [7] to complete the proof. If f is a function, by ran f we mean the range of f. Corollary 3. Suppose that h is a nonconstant multiplier of H such that ran h intersects the unit circle. Then the adjoint of the multiplication operator Mh satisfies the Hypercyclicity Criterion. Proof. In Theorem 2, let ϕ be identity and w1 = w2 = h. Now the sets Fm = {λ ∈ D : |h(λ)|m < 1} are nonempty and open sets in D and so clearly have limit points in D for m = −1, 1. But Fm ⊂ Em where Em = {λ ∈ D : ∞ i=1 h(λ)2m = 0} for m = −1, 1. Now we can apply the result of Theorem 2, and so the proof is complete. References [1] J. Bes, Three problems on hypercyclic operators, PhD Thesis, Kent State University (1998). [2] J. Bes, and A. Peris, Hereditarily hypercyclic operators, Jour. Func. Anal., 167, No. 1 (1999), 94-112. [3] P.S. Bourdon, Orbits of hyponormal operators, Mich. Math. Jour., 44 (1997), 345-353. [4] P.S. Bourdon and J.H. Shapiro, Cyclic phenomena for composition opera- tors, Memoirs of the Amer. Math. Soc., 125, AMS, Prov. - RI, 1997. [5] P.S. Bourdon and J.H. Shapiro, Hypercyclic operators that commute with the Bergman backward shift, Trans. Amer. Math. Soc., 352, No. 11 (2000), 5293-5316.
- 5. HYPERCYCLICITY CRITERION FOR A PAIR OF... 219 [6] R.M. Gethner and J.H. Shapiro, Universal vectors for operators on spaces of holomorphic functions, Proc. Amer. Math. Soc., 100 (1987), 281-288. [7] G. Godefroy, J.H. Shapiro, Operators with dense, invariant cyclic vector manifolds, Jour. of Func. Anal., 98 (1991), 229-269. [8] K. Goswin, G. Erdmann, Universal families and hypercyclic operators, Bull. of the Amer. Math. Soc., 35 (1999), 345-381. [9] D.A. Herrero, Limits of hypercyclic and supercyclic operators, Jour. of Func. Anal., 99 (1991), 179-190. [10] C. Kitai, Invariant closed sets for linear operators, Thesis, University of Toronto, 1982. [11] C. Read, The invariant subspace problem for a class of Banach spaces, 2: Hypercyclic operators, Isr. J. Math., 63 (1988), 1-40. [12] S. Rolewicz, On orbits of elements, Studia Math., 32 (1969), 17-22. [13] H.N. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc., 347 (1995), 993-1004. [14] J.H. Shapiro, Composition Operators and Classical Function Theory, Springer-Verlag, New York, 1993. [15] B. Yousefi and H. Rezaei, Hypercyclic property of weighted composition operators, Proc. Amer. Math. Soc., 135, No. 10 (2007), 3263-3271. [16] B. Yousefi and H. Rezaei, On the supercyclicity and hypercyclicity of the operator algebra, Acta Mathematica Sinica, 24, No. 7 (2008), 1221-1232. [17] B. Yousefi and R. Soltani, Hypercyclicity of the adjoint of weighted com- position operators, Proc. Indian Acad. Sci., 119, No. 3 (2009), 513-519.
- 6. 220