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 q 2 K ⁎ A p K B q 2 ) s ] 1 k , which generalizes Lieb's concavity theorem and its following results from trace to a class of homogeneous functions Tr k [ ⋅ ] 1 k . Here Tr k [ A ] denotes the k th 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.