Generalizing Lieb's Concavity Theorem via operator interpolation

Abstract

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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