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Composition operators in the Dirichlet series setting. (English) Zbl 1127.47026

Arendt, Wolfgang (ed.) et al., Perspectives in operator theory. Papers of the workshop on operator theory, Warsaw, Poland, April 19–May 3, 2004. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 75, 261-287 (2007).
This is a useful survey on composition operators acting on the Hardy-Hilbert space \(H^2\) of all analytic functions \(f(z)=\sum_{n=0}^\infty a_nz^n\) for which \(| | f| | _2:=\sum_{n=0}^\infty| a_n| ^2<\infty\), the Wiener–Banach algebra \(W^+\) of functions \(f(z)=\sum_{n=0}^\infty a_nz^n\) for which \(| | f| | _1:=\sum_{n=0}^\infty| a_n| <\infty\), the Hardy–Dirichlet Hilbert space of analytic functions \(f\) admitting a Dirichlet series expansion \(f(s)=\sum_{n=1}^\infty a_nn^{-s}\) with \(\sum_{n=1}^\infty| a_n| ^2<\infty\), and the Wiener–Dirichlet algebra \(A^+\) of analytic functions \(f\) admitting a a Dirichlet series expansion \(f(s)=\sum_{n=1}^\infty a_nn^{-s}\) with \(\sum_{n=1}^\infty| a_n| <\infty\).
The author studies under which conditions on the symbol \(\phi\), the map \(f\mapsto f\circ\phi\) is a bounded, compact, automorphic or isometric operator on these spaces. Detailed proofs are given for those results that have been hitherto only sketched in the literature. This paper is to be recommended to anyone working in this domain, including graduate students.
For the entire collection see [Zbl 1112.47300].

MSC:

47B33 Linear composition operators
30B50 Dirichlet series, exponential series and other series in one complex variable
42B35 Function spaces arising in harmonic analysis
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
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