Approximation, Gelfand, and Kolmogorov numbers of Schatten class embeddings

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 approximation quantities 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 $q$, showing how approximation numbers are intimately related to the Gelfand numbers and their duals, the Kolmogorov numbers. In particular, we obtain new bounds for those sequences of $s$-numbers. Our results improve and complement bounds previously obtained by B. Carl and A. Defant [J. Approx. Theory, 88(2):228--256, 1997], Y. Gordon, H. König, and C. Schütt [J. Approx. Theory, 49(3):219--239, 1987], A. Hinrichs and C. Michels [Rend. Circ. Mat. Palermo (2) Suppl., (76):395--411, 2005], and A. Hinrichs, J. Prochno, and J. Vybíral [preprint, 2020]. We also treat the case of quasi-Schatten norms, which is relevant in applications such as low-rank matrix recovery.

30 pages, 2 figues

References in corpus (2)

• Hastings' additivity counterexample via Dvoretzky's theorem
• Stability of low-rank matrix recovery and its connections to Banach space geometry