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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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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.
In this paper we will study the asymptotic behaviour of certain widths of the embeddings $ mathcal{A}_omega(mathbb{T}^d) to L_p(mathbb{T}^d)$, $2le p le infty$, and $ mathcal{A}_omega(mathbb{T}^d) to mathcal{A}(mathbb{T}^d)$, where $mathcal{A}_{omega}(mathbb{T}^d)$ is the weighted Wiener class and $mathcal{A}(mathbb{T}^d)$ is the Wiener algebra on the $d$-dimensional torus $mathbb{T}^d$. Our main interest will consist in the calculation of the associated asymptotic constant. As one of the consequences we also obtain the asymptotic constant related to the embedding $id: C^m_{rm mix}(mathbb{T}^d) to L_2(mathbb{T}^d)$ for Weyl and Bernstein numbers.
We establish the necessary and sufficient conditions for those symbols $b$ on the Heisenberg group $mathbb H^{n}$ for which the commutator with the Riesz transform is of Schatten class. Our main result generalises classical results of Peller, Janson--Wolff and Rochberg--Semmes, which address the same question in the Euclidean setting. Moreover, the approach that we develop bypasses the use of Fourier analysis, and can be applied to characterise that the commutator is of the Schatten class in other settings beyond Euclidean.
We study the embedding $text{id}: ell_p^b(ell_q^d) to ell_r^b(ell_u^d)$ and prove matching bounds for the entropy numbers $e_k(text{id})$ provided that $0<p<rleq infty$ and $0<qleq uleq infty$. Based on this finding, we establish optimal dimension-free asymptotic rates for the entropy numbers of embeddings of Besov and Triebel-Lizorkin spaces of small dominating mixed smoothness which settles an open question in the literature. Both results rely on a novel covering construction recently found by Edmunds and Netrusov.
The Thompson metric provides key geometric insights in the study or non-linear matrix equations and in many optimization problems. However, knowing that an approximate solution is within d_T units of the actual solution in the Thompson metric provides little insight into how good the approximation is as a matrix or vector approximation. That is, bounding the Thompson metric between an approximate and accurate solution to a problem does not provide obvious bounds either for the spectral or the Frobenius norm, both Schatten norms, of the difference between the approximation and accurate solution. This paper reports an upper bound on the Schatten norm of X - Y related to both the Thompson metric between X and Y and the maximum of their Schatten norms. This paper reports a similar but slightly tighter bound for the Frobenius norm of X - Y.
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