Localized Frames and Compactness

TL;DR

This paper introduces weak-localization for generalized frames, defines a class of weakly localized operators, and characterizes their boundedness and compactness, including many important operators like Toeplitz and Anti-Wick operators.

Contribution

It develops the concept of weak-localization for frames and characterizes the compactness of associated operators via Berezin transform behavior.

Findings

Characterization of compactness for weakly localized operators

Inclusion of important operators like Toeplitz and Anti-Wick operators

Boundedness criteria for weakly localized operators

Abstract

We introduce the concept of weak-localization for generalized frames and use this concept to define a class of weakly localized operators. This class contains many important operators, including: Short Time Fourier Transform multipliers, Calderon-Toeplitz operators, Toeplitz operators on various functions spaces, Anti-Wick operators, and many others. In this paper, we study the boundedness and compactness of weakly localized operators. In particular, we provide a characterization of compactness for weakly localized operators in terms of the behavior of their Berezin transform.