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Gelfand numbers of embeddings of Schatten classes

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Let \(0<p,q\le \infty \) and denote by \(\mathscr {S}_p^N\) and \(\mathscr {S}_q^N\) the corresponding Schatten classes of real \(N\times N\) matrices. We study the Gelfand numbers of natural identities \(\mathscr {S}_p^N\hookrightarrow \mathscr {S}_q^N\) 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. Chávez-Domínguez and D. Kutzarova.

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Acknowledgements

We thank Michał Strzelecki for comments on a preliminary version of this paper. We also gratefully acknowledge the support of the Oberwolfach Research Institute for Mathematics, where several discussions about this problem were held during the workshop “New Perspectives and Computational Challenges in High Dimensions” (Workshop ID 2006b).

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Correspondence to Joscha Prochno.

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A. Hinrichs and J. Prochno are supported by Project F5513-N26 of the Austrian Science Fund (FWF), which is a part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”. J. Prochno is also supported by the Austrian Science Fund (FWF) Project P32405 “Asymptotic geometric analysis and applications”. The research of J. Vybíral was supported by the grant P201/18/00580S of the Grant Agency of the Czech Republic and by the European Regional Development Fund-Project “Center for Advanced Applied Science” (No. CZ.02.1.01/0.0/0.0/16_019/0000778).

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Hinrichs, A., Prochno, J. & Vybíral, J. Gelfand numbers of embeddings of Schatten classes. Math. Ann. 380, 1563–1593 (2021). https://doi.org/10.1007/s00208-021-02203-9

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