Compact composition operators on a Hilbert space of Dirichlet series
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 \(\mathbb{C}_{(1/2)}:= \{ s \in \mathbb{C} \;; \;Re(s)> 1/2 \}\). \textit{J. Gordon} and \textit{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 \textit{J. McCarthy} [Trans. Am. Math. Soc. 356, 881--893 (2004; Zbl 1039.30001)].
scientific article; zbMATH DE number 1993657