Time-Frequency Analysis and Coorbit Spaces of Operators

TL;DR

This paper develops an operator-valued Short-Time Fourier Transform to analyze operators, establishing coorbit spaces of operators with properties analogous to function spaces, and classifies operators generating equivalent norms.

Contribution

It introduces a novel operator-valued transform and coorbit spaces for operators, extending time-frequency analysis to operator theory with classification and atomic decomposition results.

Findings

Operators generating equivalent norms are fully classified.

Operator spaces share atomic decomposition properties with function spaces.

Characterization of operator spaces using localisation operators.

Abstract

We introduce an operator valued Short-Time Fourier Transform for certain classes of operators with operator windows, and show that the transform acts in an analogous way to the Short-Time Fourier Transform for functions, in particular giving rise to a family of vector-valued reproducing kernel Banach spaces, the so called coorbit spaces, as spaces of operators. As a result of this structure the operators generating equivalent norms on the function modulation spaces are fully classified. We show that these operator spaces have the same atomic decomposition properties as the function spaces, and use this to give a characterisation of the spaces using localisation operators.