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Compact differences of composition operators. (English) Zbl 1087.47032
Let \(\varphi\) be an analytic map from the open unit disk \(D\) to itself. Then \(\varphi\) induces a linear operator \(C_{\varphi}\) defined by the formula \(C_{\varphi}f(z)=f\circ \varphi(z)\). Over the past thirty years, much effort has been expended on characterizing those analytic maps \(\varphi\) which induce composition operators acting on functional Hilbert spaces such as the Hardy space and the weighted Bergman spaces (see the references listed in the paper under review). Based on B. MacCluer, S. Ohno and R. Zhao’s work [Integral Equations Oper. Theory 40, No. 4, 481–494 (2001; Zbl 1062.47511)], this paper deals with compact differences of composition operators acting on the weighted Bergman spaces \(A^2_{\alpha}\), \(\alpha >-1\).
Using the pseudo-hyperbolic metric, the author gives a characterization of compact differences on \(A^2_{\alpha}\) and necessary conditions on a large class of weighted Dirichlet spaces. Conditions are also given under which a composition operator on \(A^2_{\alpha}\) can be written as a finite sum of composition operators modulo the compact ones. The additive structure of the space of composition operators modulo the compact operators is investigated further and a sufficient condition is given to insure that two composition operators lie in the same component.

MSC:
47B33 Linear composition operators
46E15 Banach spaces of continuous, differentiable or analytic functions
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