Entropy of Compact Operators with Applications to Landau-Pollak-Slepian Theory and Sobolev Spaces

TL;DR

This paper establishes a precise relation between the entropy of compact operators and their eigenvalues, applying this to improve characterizations in Landau-Pollak-Slepian theory and Sobolev spaces.

Contribution

It introduces a new general relation linking operator entropy to eigenvalues and enhances existing characterizations in specific mathematical contexts.

Findings

Derived a general relation between operator entropy and eigenvalues

Improved characterizations of Landau-Pollak-Slepian operator entropy

Enhanced understanding of Sobolev space metric entropy

Abstract

We derive a precise general relation between the entropy of a compact operator and its eigenvalues. It is then shown how this result along with the underlying philosophy can be applied to improve substantially on the best known characterizations of the entropy of the Landau-Pollak-Slepian operator and the metric entropy of unit balls in Sobolev spaces.