Schatten类嵌入的逼近、Gelfand数和Kolmogorov数
Approximation, Gelfand, and Kolmogorov numbers of Schatten class embeddings
摘要：
设$0<p,q\leq \infty$，用$\mathcal S_p^N$和$\mathcal S_q^N$表示实$N\times N$矩阵的相应Schatten类。我们研究了Schatten类之间的自然恒等式$\mathcal S_p^N\hookrightarrow \mathcal S_q^N$的逼近量，证明了渐近尖锐的界限，常数只取决于$p$和$q$，显示如何逼近数是密切相关的Gelfand数和它们的乘积，Kolmogorov数。特别地，我们得到了这些序列的$s$ -号码的新的界限。我们的结果改进和补充了B. Carl和A. Defant [J. Approx. Theory，88（2）：228- 256，1997]，Y. Gordon，H. König和C. Schütt [J. Approx. Theory，49（3）：219- 239，1987]，A. Hinrichs和C. Michels [Rend. Circ. Mat.巴勒莫（2）补充，(76)：395- 411，2005]和A. Hinrichs，J. Prochno和J. Vybíral [预印本，2020]。我们还讨论了准Schatten范数的情况，这与低秩矩阵恢复等应用相关。
英文摘要：
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. 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.
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