Approximation, Gelfand, and Kolmogorov numbers of Schatten class embeddings
share
Let 0<p,q≤∞ and denote by 𝒮_p^N and 𝒮_q^N the corresponding Schatten classes of real N× N matrices. We study approximation quantities of natural identities 𝒮_p^N↪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.
READ FULL TEXT