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Weighted composition operators from Zygmund spaces into Bloch spaces. (English) Zbl 1215.47022

Let 𝔻={z:|z|<1} be the open unit disk in the complex plane and H(𝔻) the class of all analytic functions on 𝔻, and let ϕ be an analytic self-map, that is, ϕ:𝔻𝔻. Then ϕ induces the composition operator C ϕ on H(𝔻), defined by (C ϕ )(z)=f(ϕ(z)) for z𝔻 and fH(𝔻). Let u be a fixed analytic function on 𝔻. The weighted composition operator uC ϕ is defined on H(𝔻) by ((uC ϕ )f)(z)=u(z)f(ϕ(z)), z𝔻, fH(𝔻)·

Let 𝒵 denote the space of all fH(𝔻)C(𝔻 ¯) such that

f 𝒵 =sup|f(e i(θ+h)) +f(e i(θ-h) )-2f(e iθ )| h<,

where the supremum is taken over all e iθ 𝔻 and h>0. The little Zygmund space 𝒵 0 is defined by f𝒵 0 if and only if lim |z|1 (1-|z|)|f '' (z)|=0. An analytic function f is said to belong to the Bloch space B if B(f)=sup z𝔻 (1-|z| 2 )|f ' (z)|<. Finally, the closed subspace B 0 (the little Bloch space) of B consists of the functions fB satisfying lim |z|1 (1-|z| 2 )|f ' (z)|=0. Let X 1 and X 2 be the Banach spaces. For a given operator A:X 1 X 2 , the problem of investigating its boundedness and compactness is a natural and important problem of operator theory.

In this paper, the authors study the boundedness and compactness of weighted composition operators from Zygmund space into Bloch space and little Bloch space, based mainly on the classical methods of function theory.

47B33Composition operators
46E15Banach spaces of continuous, differentiable or analytic functions
30H05Bounded analytic functions