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Linear relations in the Calkin algebra for composition operators. (English) Zbl 1115.47023

The purpose of this interesting paper is to consider the following problem: When is a finite linear combination of composition operators, acting on the Hardy space or on the standard weighted Bergman spaces on the unit disc, a compact operator? The set of composition operators, as a subset of the algebra of all bounded operators, has no obvious additive or linear structure. Nevertheless, J. Moorhouse [J. Funct. Anal. 219, No. 1, 70–92 (2005; Zbl 1087.47032)] observed additive structure in the weighted Bergman spaces, modulo the ideal of compact operators, and characterized the pairs of analytic self-maps \(\varphi\) and \(\psi\) on the unit disc such that the difference \(C_{\varphi} - C_{\psi}\) is compact.
In the present paper, analogous results for the Hardy space \(H^2\) are obtained, and the authors are able to pass from additive to linear structure in the Calkin algebra. The problem of compact difference of composition operators was raised by Shapiro and Sundberg and by MacCluer in 1989–90. The pseudo-hyperbolic distance between the values of \(\varphi\) and \(\psi\) played some role. These results are related to the characterizations of those \(\varphi\) and \(\psi\) such that the corresponding composition operators lie in the same connected component of the topological space of composition operators endowed with the operator norm. In order to obtain several results due to Moorhouse [loc. cit.] for \(H^2\), different methods are needed. An application of Clark measures is essential for the authors.
In Section 3, they obtain essential norm estimates for weighted composition operators on \(H^2\) similar to the Cima–Matheson essential norm formula for unweighted composition operators. The \(H^2\) results are presented in Section 4. Section 5 is devoted to the question of when a finite linear combination of composition operators is compact. The authors further develop ideas of MacCluer and obtain lower bounds of the essential norm in terms of first and higher order boundary data. A special class \(S\) of analytic self-maps on the unit disc, which have sufficient data at every point of the boundary at which the map makes significant contact with the boundary, is introduced. For pairs of analytic self-maps in the class \(S\), the obstructions to the essential norm of the difference being small are clarified completely, as well as conditions under which it must be small or 0.
The question mentioned at the beginning of this review is then completely solved for finite combinations of composition operators with symbols in the class \(S\). In the final Section 6, in the case when the symbol \(\varphi\) lies in a certain subclass of the class \(S\), the authors determine when another composition operator \(C_{\psi}\) belongs to the same component as \(C_{\varphi}\) both for Hardy and weighted Bergman spaces.

MSC:

47B33 Linear composition operators
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
30D55 \(H^p\)-classes (MSC2000)
47C05 Linear operators in algebras
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