Gelfand numbers of embeddings of Schatten classes

Abstract

Let $0<p,q\le \infty$ 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. Chávez-Domínguez and D. Kutzarova.

25 pages, 2 figues