Fredholm equations for non-symmetric kernels, with applications to iterated integral operators

TL;DR

This paper develops the Jordan form and SVD for non-symmetric kernel integral operators, enabling solutions to Fredholm equations and analysis of operator powers, with applications to iterated integral operators.

Contribution

It introduces methods to analyze non-symmetric kernel operators using Jordan form and SVD, advancing solutions to Fredholm equations and understanding operator behavior.

Findings

Derived Jordan form and SVD for non-symmetric kernels

Provided solutions for Fredholm equations with non-symmetric kernels

Analyzed the asymptotic behavior of operator powers

Abstract

We give the Jordan form and the Singular Value Decomposition for an integral operator ${\cal N}$ with a non-symmetric kernel $N(y,z)$. This is used to give solutions of Fredholm equations for non-symmetric kernels, and to determine the behaviour of ${\cal N}^n$ and $({\cal N}{\cal N^*})^n$ for large $n$.