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Compact composition operators on a Hilbert space of Dirichlet series. (English) Zbl 1059.47023

Let \(H\) be the Hilbert space of Dirichlet series \(f(s)= \sum_{n=1}^{\infty} a_n n^{-s}\) with square summable coefficients. The functions in \(H\) are holomorphic on the half plane \(\mathbb{C}_{(1/2)}:= \{ s \in \mathbb{C} \;; \;Re(s)> 1/2 \}\). J. Gordon and H. Hedenmalm [Mich. Math. J. 46, 313–329 (1999; Zbl 0963.47021)] characterized the analytic maps \(\phi: \mathbb{C}_{(1/2)} \rightarrow \mathbb{C}_{(1/2)}\) such that the composition operator \(C_{\phi}(f)=f \circ \phi\) is bounded on \(H\). The purpose of the present paper is to present sufficient conditions and necessary conditions for \(C_{\phi}\) to be compact on \(H\). The spectrum of the operator \(C_{\phi}\) is described in the case that a power of \(C_{\phi}\) is compact. The point spectrum of \(C_{\phi}\) is also investigated. In the last section, the main result about a sufficient condition for \(C_{\phi}\) to be compact is extended to certain weighted Hilbert spaces of Dirichlet series recently introduced by J. McCarthy [Trans. Am. Math. Soc. 356, 881–893 (2004; Zbl 1039.30001)].

MSC:

47B33 Linear composition operators
47B07 Linear operators defined by compactness properties
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