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Generalizing Lieb's Concavity Theorem via operator interpolation

Huang, De (2020) Generalizing Lieb's Concavity Theorem via operator interpolation. Advances in Mathematics, 369 . Art. No. 107208. ISSN 0001-8708. doi:10.1016/j.aim.2020.107208.

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We introduce the notion of k-trace and use interpolation of operators to prove the joint concavity of the function (A,B)↦Tr_k[(B^(qs/2)K∗A^(ps)KB^(qs/2))1/s]1/k, which generalizes Lieb's concavity theorem from trace to a class of homogeneous functions Tr_k[⋅]1/k. Here Tr_k[A] denotes the kth elementary symmetric polynomial of the eigenvalues of A. This result gives an alternative proof for the concavity of A↦Tr_k[exp(H+log A)]1/k that was obtained and used in a recent work to derive expectation estimates and tail bounds on partial spectral sums of random matrices.

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Additional Information:© 2020 Elsevier Inc. Received 5 April 2019, Revised 18 February 2020, Accepted 30 April 2020, Available online 13 May 2020. The research was in part supported by the NSF Grant DMS-1613861. The author would like to thank Thomas Y. Hou for his wholehearted mentoring and supporting.
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Subject Keywords:Trace inequalities; Concave/convex matrix functions; Interpolation of operators
Classification Code:MSC: 47A57; 47A63; 15A42; 15A16; 15A75
Record Number:CaltechAUTHORS:20200514-131620225
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Official Citation:De Huang, Generalizing Lieb's Concavity Theorem via operator interpolation, Advances in Mathematics, Volume 369, 2020, 107208, ISSN 0001-8708, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:103202
Deposited By: Tony Diaz
Deposited On:14 May 2020 21:05
Last Modified:16 Nov 2021 18:19

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