A note on numerical singular values of compositions with non-compact operators

TL;DR

This paper investigates the numerical singular values of compositions involving non-compact operators, providing numerical evidence and explanations for their decay rates, which clarifies previous ambiguities in the field.

Contribution

It supplies the missing numerical results and theoretical explanations for the singular value decay of composed operators involving non-compact and compact operators.

Findings

Numerical singular values decay exponentially fast.

Theoretical explanation reconciles previous conflicting results.

Provides insights into the behavior of non-compact operator compositions.

Abstract

Linear non-compact operators are difficult to study because they do not exist in the finite dimensional world. Recently, Math\'{e} and Hofmann studied the singular values of the compact composition of the non-compact Hausdorff moment operator and the compact integral operator and found credible arguments, but no strict proof, that those singular values fall only slightly faster than those of the integral operator alone. However, the fact that numerically the singular values of the combined operator fall exponentially fast was not mentioned. In this note, we provide the missing numerical results and provide an explanation why the two seemingly contradicting results may both be true.