Multipliers on Hilbert Spaces of Dirichlet Series

TL;DR

This paper investigates the structure of multiplier algebras on Hilbert spaces of Dirichlet series with various weighted norms, extending previous classifications and exploring new weight classes including multiplicative and number-theoretic sequences.

Contribution

It extends the classification of multipliers on Hilbert spaces of Dirichlet series to new weight classes, including multiplicative weights and number-theoretic sequences, with bounds on operator norms.

Findings

Derived bounds on operator norms of multipliers.

Classified multiplier algebras for new weight sequences.

Provided examples matching McCarthy's results under different conditions.

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.