1. International Journal of Applied Mathematics
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Volume 24 No. 4 2011, 625-629
ON THE EIGHENFUNCTIONS OF HYPERBOLIC WEIGHTED
COMPOSITION OPERATORS ON FUNCTION SPACES
B. Yousefi1 §, M. Habibi2
1Department of Mathematics
Payame Noor University
P.O. Box 71955-1368, Shiraz, IRAN
e-mails: b yousefi@pnu.ac.ir, bahmann@spnu.ac.ir
2Department of Mathematics
Branch of Dehdasht
Islaamic Azad University
P.O. Box 7571763111, Dehdasht, IRAN
e-mail: habibi.m@iaudehdasht.ac.ir
Abstract: In this paper we characterize the eigenfunctions of certain weighted
composition operators Cϕ,ψ acting on Hilbert spaces of analytic functions where
ψ is of hyperbolic type and ϕ is nonzero on the Denjoy-Wolff point of ψ.
AMS Subject Classification: 47B37, 47A25
Key Words: Hilbert spaces of analytic functions, reproducing kernels, weighted
composition operator, Farrell-Rubel-Shields theorem
1. Introduction
Let H be a Hilbert space of analytic functions on the open unit disk U such that
for each z ∈ U, the evaluation function eλ : H → C defined by eλ(f) = f(λ)
is bounded on H. By the Riesz Representation Theorem there is a vector
kz ∈ H such that f(z) =< f, kz > for every z ∈ U. Furthermore, we assume
that H contains the constant functions and multiplication by the independent
variable z defines a bounded linear operator Mz on H. The operator Mz is
called polynomially bounded on H if there exists a constant d > 0 such that
Received: May 24, 2011 c 2011 Academic Publications
§Correspondence author

2. 626 B. Yousefi, M. Habibi
||Mp|| ≤ d||p||U for every polynomial p. Here ||p||U denotes the supremum norm
of p on U. It is well-known that any operator which is similar to a contraction is
polynomially bounded. A complex valued function ϕ on U for which ϕH ⊆ H
is called multiplier of H. The set of all multipliers of H is denoted by M(H)
and it is well-known that M(H) ⊆ H∞(U) ([7]). Moreover, we suppose that ψ
is a holomorphic self-map of U that is not an elliptic automorphism, and ϕ is a
nonzero multiplier of H which is defined as radial limit at the Denjoy-Wolff point
of ψ. The weighted composition operator Cϕ,ψ acting on H is defined by Cϕ,ψ =
MϕCψ. The adjoint of a composition operator and a multiplication operator
has not been yet well characterized on holomorphic functions. Nevertheless
their action on reproducing kernels is determined. In fact C∗
ψ(kz) = kψ(z) and
M∗
ϕ(kz) = ϕ(z)kz for every z ∈ U. Thus for each f in H and z ∈ U, we have
C∗
ϕ,ψkz = C∗
ψM∗
ϕkz = ϕ(z)C∗
ψkz = ϕ(z)kψ(z).
For some sources related to the topics of this paper one can see [1]-[8].
2. Main Result
In the main theorem of this paper we characterize the eigenfunctions of weighted
composition operators acting on Hilbert spaces of analytic functions. The holo-
morphic self maps of the open unit disc U are divided into classes of elliptic and
non-elliptic. The elliptic type is an automorphism and has a fixed point in U.
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
([6]). By ψn we denote the nth iterate of ψ and by ψ (w) we denote the angular
derivative of ψ at w ∈ ∂U. Note that if w ∈ U, then ψ (w) has the natural
meaning of derivative.
Theorem 1. (Denjoy-Wolff Iteration Theorem) Suppose ψ is a holomorphic
self-map of U that is not an elliptic automorphism.
(i) If ψ has a fixed point w ∈ U, then ψn → w uniformly on compact
subsets of U, and |ψ (w)| < 1.
(ii) If ψ has no fixed point in U, then there is a point w ∈ ∂U such that
ψn → w uniformly on compact subsets of U, and the angular derivative of ψ
exists at w, with 0 < ψ (w) ≤ 1.
We call the unique attracting point w, the Denjoy-Wolff point of ψ. By
the Denjoy-Wolff Iteration Theorem, a general classification for non-elliptic

3. ON THE EIGHENFUNCTIONS OF HYPERBOLIC WEIGHTED... 627
holomorphic self maps of U can be given: let w be the Denjoy-Wolff point of a
holomorphic self-map ψ of U. We say ψ is of dilation type if w ∈ U, of hyperbolic
type if w ∈ ∂U and ψ (w) < 1, and of parabolic type if w ∈ ∂U and ψ (w) = 1.
Theorem 2. Let w be the Denjoy-Wolff fixed point of ψ which is of hyper-
bolic type and ϕ(w) = 0. If Mz is polynomially bounded and ϕ◦ψn U ≤ |ϕ(w)|
for all n, then
∞
n=0
1
ϕ(w)
ϕ ◦ ψn
is an eigenfunction of the weighted composition operator Cϕ,ψ acting on H.
Proof. Let w be the Denjoy-Wolff fixed point of ψ. By the Denjoy-Wolff
Iteration Theorem, the angular derivative of ψ exists at w, and 0 < ψ (w) < 1.
Setting r = ψ (w), it follows from the Julia-Caratheodory Inequality ([6]) that
|ψ(z) − w|2
1 − |ψ(z)|2
< r
|z − w|2
1 − |z|2
.
By substituting ψn(z) for z, in the above inequality we get
|ψn(z) − w|2
1 − |ψn(z)|2
< rn |z − w|2
1 − |z|2
for every z ∈ U and for all nonnegative integer n. Now if K is a compact subset
of U, then there exists a constant M > 0 such that
|z − w|2
1 − |z|2
< M for all z in
K. So it follows that
1 − |ψn(z)| ≤ |ψn(z) − w|
≤
|ψn(z) − w|2
1 − |ψn(z)|2
≤ rn |z − w|2
1 − |z|2
< Mrn
.
Thus
∞
i=0
(1 − |ψi(z)|) converges uniformly on compact subsets of U. Note that
since ϕ is differentiable in w, there exist some constant c1 and δ > 0, such that
|ϕ(w) − ϕ(z)| < c1|w − z|

4. 628 B. Yousefi, M. Habibi
for every z with |z − w| < δ. Now fix a compact subset K of U. Now the Julia-
Caratheodory Inequality (see [3, Theorem 1.3]) provides a positive integer N
and a constant c > 0 such that
|w − ψn(z)| < c(1 − |ψn(z)|)
for each z ∈ K and every n > N. In the first relation by substituting ψn(z)
instead of z we get
|ϕ(w) − ϕ(ψn(z))| < c c1(1 − |ψn(z)|)
for every n > N. So
|1 −
1
ϕ(w)
ϕ(ψn(z))| <
cc1
ϕ(w)
(1 − |ψn(z)|).
As we saw,
∞
n=0
(1 − |ψn(z)|) and consequently
∞
n=0
|1 −
1
ϕ(w)
ϕ(ψn(z))|
converges uniformly on K. Thus
∞
n=0
1
ϕ(w)
ϕ(ψn(z))
also converges uniformly on K. Define
g(z) =
∞
n=0
1
ϕ(w)
ϕ(ψn(z)).
Thus g is a holomorphic function on U. If g(z) = 0 for some z ∈ U,then
ϕ(ψi(z)) = 0 for some integer i. If g is the constant function 0, then U is a
subset of
∞
i=0
Ai, where
Ai = {z ∈ U : ϕ(ψi(z)) = 0}
for every integer i. The Bair-Category Theorem implies that Ai has nonempty
interior for some integer i, so ϕ must be identically zero. This is a contradiction.
Thus g is a nonzero holomorphic function on U and
ϕ(z) · g(ψ(z)) = ϕ(w)g(z).

5. ON THE EIGHENFUNCTIONS OF HYPERBOLIC WEIGHTED... 629
Since ϕ◦ψn U ≤ |ϕ(w)| for all n, thus g belongs to H∞(U). Now by the Farrell-
Rubel-Shields Theorem (see [3, Theorem 5.1, p. 151]), there is a sequence {pn}n
of polynomials converging to g such that for all n, pn U ≤ c0 for some c0 > 0.
So we obtain
Mpn ≤ d pn U ≤ dc0
for all n. But ball B(H) is compact in the weak operator topology and so by
passing to a subsequence if necessary, we may assume that for some A ∈ B(H),
Mpn −→ A in the weak operator topology. Using the fact that M∗
pn
−→ A∗ in
the weak operator topology and acting these operators on eλ we obtain that
pn(λ)eλ = M∗
pn
eλ −→ A∗
eλ
weakly. Since pn(λ) −→ g(λ) we see that A∗eλ = g(λ)eλ. Because the closed
linear span of {kλ : λ ∈ U} is dense in H, we conclude that A = Mg and this
implies that g ∈ M(H). Hence indeed g ∈ H, since H contains the constant
functions. This completes the proof.
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