Approximation, Gelfand, and Kolmogorov numbers of Schatten class embeddings

Abstract

Let 0<p,q≤∞ and denote by SpN and SqN the corresponding Schatten classes of real N×N matrices. We study approximation quantities of natural identities SpN↪SqN 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 Carl and Defant (1997), Gordon et al. (1987), Hinrichs and Michels (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.