Abstract
In this paper we will show how the boundedness condition for the weighted composition operators on a class of spaces of analytic functions on the open right complex half-plane called Zen spaces (which include the Hardy spaces and weighted Bergman spaces) can be stated in terms of Carleson measures and Bergman kernels. In Hilbertian setting we will also show how the norms of causal weighted composition operators on these spaces are related to each other and use it to show that an (unweighted) composition operatorC_varphi is bounded on a Zen space if and only if varphi has a finite angular derivative at infinity. Finally, we will show that there is no compact composition operator on Zen spaces.
Highlights
Let L be a linear space of complex-valued functions defined on a domain ⊆ C, let φ be a self map of, and let h : → C
[6]) that all unweighted composition operators are bounded on all Hardy spaces and weighted Bergman spaces
Some further results concerning so-called Zen spaces and weighted composition operators have been obtained in [3] and we aim to extend it in this article
Summary
Let L be a linear space of complex-valued functions defined on a domain ⊆ C, let φ be a self map of , and let h : → C. The particular case of weighted and unweighted composition operators on spaces of analytic functions defined on = D d=efn{z ∈ C : |z| < 1}, the open unit disk of the complex plane, have been studied very extensively. It follows from the Littlewood subordination principle (see [6]) that all unweighted composition operators are bounded on all Hardy spaces and weighted Bergman spaces (see [5]). Wynn in [7] (for weighted Bergman spaces) They have found the expression for the norm of composition operators for these spaces and shown that the composition operators must never be compact in this context. Throughout this paper h is always defined to be a holomorphic map on C+ and φ a holomorphic self-map on C+
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