International Journal of Applied Mathematics————————————————————–Volume 24 No. 4 2011, 625-629

ON THE EIGHENFUNCTIONS OF HYPERBOLIC WEIGHTED

COMPOSITION OPERATORS ON FUNCTION SPACES

B. Yousefi1 §, M. Habibi2

1Department of MathematicsPayame Noor University

P.O. Box 71955-1368, Shiraz, IRANe-mails: b yousefi@pnu.ac.ir, bahmann@spnu.ac.ir

2Department of MathematicsBranch of Dehdasht

Islaamic Azad UniversityP.O. Box 7571763111, Dehdasht, IRAN

e-mail: habibi.m@iaudehdasht.ac.ir

Abstract: In this paper we characterize the eigenfunctions of certain weightedcomposition operators Cϕ,ψ acting on Hilbert spaces of analytic functions whereψ is of hyperbolic type and ϕ is nonzero on the Denjoy-Wolff point of ψ.

AMS Subject Classification: 47B37, 47A25Key Words: Hilbert spaces of analytic functions, reproducing kernels, weightedcomposition operator, Farrell-Rubel-Shields theorem

1. Introduction

Let H be a Hilbert space of analytic functions on the open unit disk U such thatfor 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 vectorkz ∈ H such that f(z) =< f, kz > for every z ∈ U . Furthermore, we assumethat H contains the constant functions and multiplication by the independentvariable z defines a bounded linear operator Mz on H. The operator Mz iscalled polynomially bounded on H if there exists a constant d > 0 such that

Received: May 24, 2011 c© 2011 Academic Publications§Correspondence author

626 B. Yousefi, M. Habibi

||Mp|| ≤ d||p||U for every polynomial p. Here ||p||U denotes the supremum normof p on U . It is well-known that any operator which is similar to a contraction ispolynomially bounded. A complex valued function ϕ on U for which ϕH ⊆ His 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 ψis a holomorphic self-map of U that is not an elliptic automorphism, and ϕ is anonzero multiplier ofH which is defined as radial limit at the Denjoy-Wolff pointof 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 operatorhas not been yet well characterized on holomorphic functions. Neverthelesstheir action on reproducing kernels is determined. In fact C∗

ψ(kz) = kψ(z) andM∗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 weightedcomposition 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 andnon-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 ofa 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 angularderivative of ψ at w ∈ ∂U . Note that if w ∈ U , then ψ′(w) has the naturalmeaning of derivative.

Theorem 1. (Denjoy-Wolff Iteration Theorem) Supposeψ is a holomorphicself-map of U that is not an elliptic automorphism.

(i) If ψ has a fixed point w ∈ U , then ψn → w uniformly on compactsubsets of U , and |ψ′(w)| < 1.

(ii) If ψ has no fixed point in U , then there is a point w ∈ ∂U such thatψn → w uniformly on compact subsets of U , and the angular derivative of ψexists at w, with 0 < ψ′(w) ≤ 1.

We call the unique attracting point w, the Denjoy-Wolff point of ψ. Bythe Denjoy-Wolff Iteration Theorem, a general classification for non-elliptic

ON THE EIGHENFUNCTIONS OF HYPERBOLIC WEIGHTED... 627

holomorphic self maps of U can be given: let w be the Denjoy-Wolff point of aholomorphic self-map ψ of U . We say ψ is of dilation type if w ∈ U , of hyperbolictype 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. IfMz is polynomially bounded and ‖ϕ◦ψn‖U ≤ |ϕ(w)|for all n, then

∞∏

n=0

1ϕ(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-WolffIteration 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|21 − |ψ(z)|2 < r

|z −w|21 − |z|2 .

By substituting ψn(z) for z, in the above inequality we get

|ψn(z) − w|21 − |ψn(z)|2 < rn

|z − w|21 − |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|21 − |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|21 − |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|

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 Nand 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 − 1ϕ(w)

ϕ(ψn(z))| < cc1ϕ(w)

(1 − |ψn(z)|).

As we saw,∞∑

n=0

(1 − |ψn(z)|) and consequently

∞∑

n=0

|1 − 1ϕ(w)

ϕ(ψn(z))|

converges uniformly on K. Thus∞∏

n=0

1ϕ(w)

ϕ(ψn(z))

also converges uniformly on K. Define

g(z) =∞∏

n=0

1ϕ(w)

ϕ(ψn(z)).

Thus g is a holomorphic function on U . If g(z) = 0 for some z ∈ U ,thenϕ(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 nonemptyinterior 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).

ON THE EIGHENFUNCTIONS OF HYPERBOLIC WEIGHTED... 629

Since ‖ϕ◦ψn‖U ≤ |ϕ(w)| for all n, thus g belongs toH∞(U). Now by the Farrell-Rubel-Shields Theorem (see [3, Theorem 5.1, p. 151]), there is a sequence {pn}nof 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 bypassing 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∗pneλ −→ A∗eλ

weakly. Since pn(λ) −→ g(λ) we see that A∗eλ = g(λ)eλ. Because the closedlinear span of {kλ : λ ∈ U} is dense in H, we conclude that A = Mg and thisimplies that g ∈ M(H). Hence indeed g ∈ H, since H contains the constantfunctions. This completes the proof.

References

[1] P.S. Bourdon, J.H. Shapiro, Cyclic composition operator on H2, Proc.Symp. Pure Math., 51, No. 2 (1990), 43-53.

[2] V. Chekliar, Eigenfunctions of the hyperbolic composition operator, Integr.Equ. Oper. Theory, 29 (1997), 264-367.

[3] T. Gamelin, Uniform Algebra, New York (1984).

[4] G. Godefroy, J.H. Shapiro, Operators with dense invariant cyclic vectormanifolds, J. Func. Anal., bf 98 (1991), 229-269.

[5] V. Matache, The eigenfunctions of a certain composition operator, Con-temp. Math., 213 (1998), 121-136.

[6] J.H. Shapiro, Composition Operators and Classical Function Theory,Springer-Verlag New York (1993).

[7] A. Shields, L. Wallen, The commutant of certain Hilbert space operators,Indiana Univ. Math. J., 20 (1971), 777-788.

[8] B. Yousefi, H. Rezaei, Hypercyclic property of weighted composition oper-ators, Proc. Amer. Math. Soc., 135, No. 10 (2007), 3263-3271.

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