Generalized Bessel multipliers in Hilbert spaces

TL;DR

This paper introduces generalized Bessel multipliers in Hilbert spaces, exploring their properties, invertibility, and behavior under perturbations, with a focus on their relation to operator classes and basis types.

Contribution

It provides a comprehensive analysis of generalized Bessel multipliers, including conditions for invertibility and their behavior with Riesz bases, extending the understanding of these operators.

Findings

Multipliers associated with Riesz bases are easily invertible and composable.

Membership in operator classes depends on the symbol's class.

Conditions for invertibility and stability under perturbations are established.

Abstract

The notation of generalized Bessel multipliers is obtained by a bounded operator on $\ell^2$ which is inserted between the analysis and synthesis operators. We show that various properties of generalized multipliers are closely related to their parameters, in particular it will be shown that the membership of generalized Bessel multiplier in the certain operator classes requires that its symbol belongs in the same classes, in special sense. Also, we give some examples to illustrate our results. As we shall see, generalized multipliers associated with Riesz bases are well-behaved, more precisely in this case multipliers can be easily composed and inverted. Special attention is devoted to the study of invertible generalized multipliers. Sufficient and/or necessary conditions for invertibility are determined. Finally, the behavior of these operators under perturbations is discussed.