A note on the operator window of modulation spaces

TL;DR

This paper characterizes the window functions and operator classes that produce equivalent norms in modulation spaces via the short-time Fourier transform, with applications to localization operators.

Contribution

It provides a complete characterization of window classes and operator classes for modulation space norms, extending to operator-valued modulation spaces and applications to localization operators.

Findings

Characterization of window classes for equivalent modulation space norms

Characterization of positive bounded operators in Cohen's class distributions

Introduction of operator classes for operator-valued modulation spaces

Abstract

Inspired by a recent article \cite[JFAA, 28(2):1-34, (2022)]{Skrettingland2022JoFAaA}, this paper is devoted to the study of suitable window class in the framework of bounded linear operators on $L^2(\rd)$. We establish a natural and complete characterization for the window class such that the corresponding STFT leads to equivalent norms of modulation spaces. The positive bounded linear operators are also characterized in Cohen's class distributions such that the corresponding quantities form equivalent norms of modulation spaces. As a generalization, we introduce a family of operator classes corresponding to the operator-valued modulation spaces. Some applications of our main theorems to the localization operators are also concerned.