Approximation sequences on Banach spaces: a rich approach

TL;DR

This paper develops a new algebraic approach to analyze the stability and Fredholm properties of convolution operators on $L^p( )$, providing criteria and formulas for approximation numbers and indices.

Contribution

It introduces a simpler, more powerful algebraic technique using $ ext{P}$-theory and rich sequences to study stability and Fredholm theory of convolution-type operators.

Findings

Criteria for stability of finite sections of convolution operators.

Formulas for asymptotic approximation numbers.

Results on Fredholm indices and their behavior.

Abstract

Criteria for the stability of finite sections of a large class of convolution type operators on $L^p(\mathbb{R})$ are obtained. In this class almost all classical symbols are permitted, namely operators of multiplication with functions in $[\textrm{PC} ,\textrm{SO}, L^\infty_0]$ and convolution operators (as well as Wiener-Hopf and Hankel operators) with symbols in $[\textrm{PC},\textrm{SO},\textrm{AP},\textrm{BUC}]_p$. We use a simpler and more powerful algebraic technique than all previous works: the application of $\mathcal{P}$-theory together with the rich sequences concept and localization. Beyond stability we study Fredholm theory in sequence algebras. In particular, formulas for the asymptotic behavior of approximation numbers and Fredholm indices are given.