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تسجيل مستخدم جديدApproximation, Gelfand, and Kolmogorov numbers of Schatten class embeddings
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Let $0<p,qleq infty$ and denote by $mathcal S_p^N$ and $mathcal S_q^N$ the corresponding Schatten classes of real $Ntimes N$ matrices. We study approximation quantities of natural identities $mathcal S_p^Nhookrightarrow mathcal S_q^N$ 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. Konig, and C. Schutt [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. Vybiral [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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Let $0<p,qleq infty$ and denote by $mathcal{S}_p^N$ and $mathcal{S}_q^N$ the corresponding Schatten classes of real $Ntimes N$ matrices. We study the Gelfand numbers of natural identities $mathcal{S}_p^Nhookrightarrow mathcal{S}_q^N$ between Schatten
classes and prove asymptotically sharp bounds up to constants only depending on $p$ and $q$. This extends classical results for finite-dimensional $ell_p$ sequence spaces by E. Gluskin to the non-commutative setting and complements bounds previously obtained by B. Carl and A. Defant, A. Hinrichs and C. Michels, and J. Chavez-Dominguez and D. Kutzarova.
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Given a compact (Hausdorff) group $G$ and a closed subgroup $H$ of $G,$ in this paper we present symbolic criteria for pseudo-differential operators on compact homogeneous space $G/H$ characterizing the Schatten-von Neumann classes $S_r(L^2(G/H))$ fo
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