Unconditional convergence and invertibility of multipliers

TL;DR

This paper investigates the conditions under which multipliers, constructed from frame-like sequences and symbols, are unconditionally convergent and invertible, providing formulas for their inverses and characterizations for Riesz basis cases.

Contribution

It offers new necessary and sufficient conditions for the unconditional convergence and invertibility of multipliers, including explicit inverse formulas and Riesz basis characterizations.

Findings

Conditions for unconditional convergence are established.

Invertibility criteria depend on properties of sequences and symbols.

Explicit inverse formulas are derived for invertible multipliers.

Abstract

In the present paper the unconditional convergence and the invertibility of multipliers is investigated. Multipliers are operators created by (frame-like) analysis, multiplication by a fixed symbol, and resynthesis. Sufficient and/or necessary conditions for unconditional convergence and invertibility are determined depending on the properties of the analysis and synthesis sequences, as well as the symbol. Examples which show that the given assertions cover different classes of multipliers are given. If a multiplier is invertible, a formula for the inverse operator is determined. The case when one of the sequences is a Riesz basis is completely characterized.