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.
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