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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 [email protected], [email protected]
2Department of MathematicsBranch of Dehdasht
Islaamic Azad UniversityP.O. Box 7571763111, Dehdasht, IRAN
e-mail: [email protected]
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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