Entropy-, Approximation- and Kolmogorov Numbers on Quasi-Banach Spaces

TL;DR

This thesis introduces and compares entropy, approximation, and Kolmogorov numbers for operators on quasi-Banach spaces, providing estimates and analyzing their connections to spectral theory.

Contribution

It extends the definitions and properties of these quantities from Banach to quasi-Banach spaces, including sharp estimates for finite-dimensional cases.

Findings

Sharp estimates for entropy numbers of identity operators between finite-dimensional $ ext{ell}_p$ spaces.

Comparison and bounds among entropy, approximation, and Kolmogorov numbers.

Connections established between these quantities and spectral inequalities like Carl's and Weyl's inequalities.

Abstract

In this bachelor's thesis we introduce three quantities for linear and bounded operators on quasi-Banach spaces which are entropy numbers, approximation numbers and Kolmogorov numbers. At first we establish the three quantities with some basic properties and try to modify known content from the Banach space case. We compare each one of them, with the corresponding other two and give estimates concerning the mean values and limits. As an example, we analyze the identity operator between finite dimensional $\ell_{p}$ spaces $\mbox{id : }\left(\ell_{p}^{n}\rightarrow\ell_{q}^{n}\right)$ for $0<p,q\leq\infty$ and give sharp estimates for entropy numbers. Furthermore we add some known estimates for approximation numbers and Kolmogorov numbers. At last we examine some renowned connections of these quantities to spectral theory on infinite dimensional Hilbert spaces, which are the inequality of Carl and the inequality of Weyl.