Remarks on j-eigenfunctions of operators

TL;DR

This paper explores conditions under which compact linear operators between Banach spaces can be represented via series of j-eigenfunctions, focusing on factorization through Hilbert spaces and properties like p-compactness.

Contribution

It introduces new insights into series representations of operators using j-eigenfunctions, especially for those with specific factorization and decay properties.

Findings

Series representation possible under certain conditions

Factorization through Hilbert spaces facilitates representation

p-compactness is a useful property for these operators

Abstract

The paper is largely concerned with the possibility of obtaining a series representation for a compact linear map $T$ acting between Banach spaces. It is known that, using the notions of $j-$eigenfunctions and $j-$% eigenvalues, such a representation is possible under certain conditions on $T$. Particular cases discussed include those in which $T$ can be factorised through a Hilbert space, or has certain $s$-numbers that are fast-decaying. The notion of $p-$compactness proves to be useful in this context; we give examples of maps that possess this property.