Multipliers of Hilbert Spaces of Analytic Functions on the Complex Half-Plane

TL;DR

This paper characterizes multipliers of Hilbert spaces of analytic functions on the right half-plane using a generalized Carleson measure, linking Laplace transforms, function spaces, and operator theory.

Contribution

It provides a full characterization of multipliers in these spaces via a generalized Carleson measure, expanding understanding of their structure and algebraic properties.

Findings

Multipliers are characterized by generalized Carleson measures.

Certain spaces are Banach algebras under specific conditions.

Necessary and sufficient conditions for these spaces to be contained within their multipliers.

Abstract

It follows, from a generalised version of Paley-Wiener theorem, that the Laplace transform is an isometry between certain spaces of weighted $L^2$ functions defined on $(0, \infty)$ and (Hilbert) spaces of analytic functions on the right complex half-plane (for example Hardy, Bergman or Dirichlet spaces). We can use this fact to investigate properties of multipliers and multiplication operators on the latter type of spaces. In this paper we present a full characterisation of multipliers in terms of a generalised concept of a Carleson measure. Under certain conditions, these spaces of analytic functions are not only Hilbert spaces but also Banach algebras, and are therefore contained within their spaces of multipliers. We provide some necessary as well as sufficient conditions for this to happen and look at its consequences.