Multipliers for Continuous Frames in Hilbert Spaces

TL;DR

This paper develops the theory of continuous frame multipliers in Hilbert spaces, extending discrete frame concepts to the continuous case and analyzing properties like compactness and Schatten class membership.

Contribution

It introduces and studies continuous frame multipliers, generalizing discrete frame multipliers and exploring their properties in the context of continuous frames.

Findings

Continuous frame multipliers are characterized and their properties analyzed.

Results on compactness and Schatten class membership are established.

Controlled and weighted frames are extended to the continuous setting.

Abstract

In this paper we examine the general theory of continuous frame multipliers in Hilbert space. These operators are a generalization of the widely used notion of (discrete) frame multipliers. Well-known examples include Anti-Wick operators, STFT multipliers or Calder\'on- Toeplitz operators. Due to the possible peculiarities of the underlying measure spaces, continuous frames do not behave quite as well as their discrete counterparts. Nonetheless, many results similar to the discrete case are proven for continuous frame multipliers as well, for instance compactness and Schatten class properties. Furthermore, the concepts of controlled and weighted frames are transferred to the continuous setting.