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 $(A,B)\mapsto {\text{Tr}}_k[(B^{\frac{qs}{2}}{K}^{\ast }{A}^{ps}K{B}^{\frac{qs}{2}}{)}^{\frac{1}{s}}{]}^{\frac{1}{k}}$, which generalizes Lieb's concavity theorem from trace to a class of homogeneous functions ${\text{Tr}}_k[\cdot {]}^{\frac{1}{k}}$. Here ${\text{Tr}}_k[A]$ denotes the $k_{\text{th}}$ elementary symmetric polynomial of the eigenvalues of $A$. This result gives an alternative proof for the concavity of $A\mapsto {\text{Tr}}_k[\exp (H+\log A){]}^{\frac{1}{k}}$ that was obtained and used in a recent work to derive expectation estimates and tail bounds on partial spectral sums of random matrices.