Approximation, Gelfand, and Kolmogorov numbers of Schatten class embeddings

TL;DR

This paper investigates the approximation, Gelfand, and Kolmogorov numbers of embeddings between Schatten classes of matrices, providing asymptotically sharp bounds and extending previous results to quasi-Schatten norms relevant for applications like low-rank matrix recovery.

Contribution

It establishes new asymptotically sharp bounds for s-numbers of Schatten class embeddings, relating approximation, Gelfand, and Kolmogorov numbers, and extends results to quasi-Schatten norms.

Findings

Derived asymptotically sharp bounds for s-numbers.

Connected approximation, Gelfand, and Kolmogorov numbers.

Extended analysis to quasi-Schatten norms.

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 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. K\"onig, and C. Sch\"utt [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. Vyb\'iral [preprint, 2020]. We also treat the case of quasi-Schatten norms, which is relevant in applications such as low-rank matrix recovery.