Multipliers on Hilbert Spaces of Dirichlet Series

Abstract

In this paper, certain classes of Hilbert spaces of Dirichlet series with weighted norms and their corresponding multiplier algebras will be explored. For a sequence $\{w_n\}_{n=n_0}^\infty $ of positive numbers, define \[\mathcal H^\textbf{w}=\left\{\sum_{n=n_0}^\infty a_nn^{-s}:\sum_{n=n_0}^\infty |a_n|^2 w_n<\infty\right\}.\] Hedenmalm, Lindqvist and Seip considered the case in which $w_n\equiv 1$ and classified the multiplier algebra of $\mathcal H^\textbf{w}$ for this space in \cite {HLS}. In \cite{M}, McCarthy classified the multipliers on $\mathcal H^\textbf{w}$ when the weights are given by \[w_n=\int_0^\infty n^{-2\sigma}d\mu(\sigma),\] where $\mu$ is a positive Radon measure with $\{0\}$ in its support and $n_0$ is the smallest positive integer for which this integral is finite. Similar results will be derived assuming the weights are multiplicative, rather than given by a measure. In particular, upper and lower bounds on the operator norms of the multipliers will be obtained, in terms of their values on certain half planes, on the Hilbert spaces resulting from these weights. Finally, some number theoretic weight sequences will be explored and the multiplier algebras of the corresponding Hilbert spaces determined up to isometric isomorphism, providing examples where the conclusion of McCarthy's result holds, but under alternate hypotheses on the weights.