Inequalities between s-numbers

TL;DR

This paper explores inequalities among various s-numbers, which generalize singular values to Banach spaces, providing an elementary proof of bounds between the smallest and largest s-numbers.

Contribution

It offers a new elementary proof of bounds between the extremal s-numbers, enhancing understanding of their relationships in Banach space operator theory.

Findings

Established bounds between smallest and largest s-numbers

Unified treatment of different s-number types including approximation and Gelfand numbers

Simplified proof technique for inequalities among s-numbers

Abstract

Singular numbers of operators between Hilbert spaces were generalized to Banach spaces by s-numbers (in the sense of Pietsch). This allows for different choices, including approximation, Gelfand, Kolmogorov and Bernstein numbers. Here, we present an elementary proof of a bound between the smallest and the largest s-number.