Gelfand numbers of embeddings of Schatten classes

TL;DR

This paper investigates the Gelfand numbers of embeddings between Schatten classes of matrices, providing sharp asymptotic bounds that extend classical sequence space results to the non-commutative matrix setting.

Contribution

It establishes asymptotically sharp bounds for Gelfand numbers of Schatten class embeddings, extending classical results from sequence spaces to matrix Schatten classes.

Findings

Derived sharp bounds for Gelfand numbers of Schatten class embeddings.

Extended classical $ ext{ell}_p$ space results to non-commutative matrix spaces.

Complemented previous bounds with new asymptotic estimates.

Abstract

Let $0<p,q\leq \infty$ and denote by $\mathcal{S}_p^N$ and $\mathcal{S}_q^N$ the corresponding Schatten classes of real $N\times N$ matrices. We study the Gelfand numbers of natural identities $\mathcal{S}_p^N\hookrightarrow \mathcal{S}_q^N$ between Schatten classes and prove asymptotically sharp bounds up to constants only depending on $p$ and $q$. This extends classical results for finite-dimensional $\ell_p$ sequence spaces by E. Gluskin to the non-commutative setting and complements bounds previously obtained by B. Carl and A. Defant, A. Hinrichs and C. Michels, and J. Ch\'avez-Dom\'inguez and D. Kutzarova.