Memory Estimation of Inverse Operators

TL;DR

This paper develops a harmonic analysis framework to estimate the memory decay of inverse operators in Banach spaces, generalizing Wiener's lemma to non-commutative settings and applying it to integral operators.

Contribution

It introduces a non-commutative version of Wiener's Tauberian lemma using Beurling spectrum to estimate inverse operator memory decay.

Findings

Generalized Wiener's lemma for non-commutative operators

Provided estimates for inverse matrix elements

Applied results to integral and integro-differential operators

Abstract

We use methods of harmonic analysis and group rep- resentation theory to estimate memory decay of the inverse oper- ators in Banach spaces. The memory of the operators is defined using the notion of the Beurling spectrum. We obtain a general continuous non-commutative version of the celebrated Wiener's Tauberian lemma with estimates of the "Fourier coefficients" of inverse operators. In particular, we generalize various estimates of the elements of the inverse matrices. The results are illus- trated with a variety of examples including integral and integro- differential operators.