Study on operator representation of frames in Hilbert spaces

TL;DR

This paper explores the operator structure of frames in Hilbert spaces, analyzing the properties and size of sets of elements associated with specific operators, and connecting frame theory with operator theory.

Contribution

It provides an overview of the operator structure of frames, investigates the size of related element sets, and raises questions linking frame and operator theories.

Findings

Characterization of operator classes associated with frames

Results on the size of sets of elements linked to these operators

Open questions connecting frame and operator theories

Abstract

The purpose of this paper is to give an overview of the operator structure of frames, where the operator belongs to certain classes of linear operators and the element belongs to $H$. We discuss the size of the set of such elements. Also, for a given frame and any, some results are obtained for . Finally, we conclude this note by raising several questions connecting frame theory and operator theory.