Characterization of continuous g-frames via operators

TL;DR

This paper explores continuous g-frames in Hilbert spaces, introducing new notions, characterizations, and decompositions involving operators, orthonormal bases, and Riesz bases, enhancing understanding of their structure.

Contribution

It introduces cg-orthonormal bases, shows how cg-frames can be represented via operators, and establishes decompositions into Parseval and Riesz bases.

Findings

Every cg-frame can be expressed as a composition of a cg-orthonormal basis and an operator.

Each cg-frame can be decomposed into two Parseval cg-frames.

Every cg-frame can be written as a sum of a cg-orthonormal basis and a cg-Riesz basis.

Abstract

In this paper we introduce and show some new notions and results on cg-frames of Hilbert spaces. We define cg-orthonormal bases for a Hilbert space H and verify their properties and relations with cg-frames. Actually, we present that every cg-frame can be represented as a composition of a cg-orthonormal basis and an operator under some conditions. Also, we find for any cg-frame an induced c-frame and study their properties and relations. Moreover, we show that every cg-frame can be written as aggregate of two Parseval cg-frames. In addition, We show each cg-frame as a summation of a cg-orthonormal basis and a cg-Riesz basis.