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 $(A,B)\mapsto\text{Tr}_k\big[(B^\frac{qs}{2}K^*A^{ps}KB^\frac{qs}{2})^{\frac{1}{s}}\big]^\frac{1}{k}$, 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 $A\mapsto\text{Tr}_k\big[\exp(H+\log A)\big]^\frac{1}{k}$ that was obtained and used in a recent work to derive expectation estimates and tail bounds on partial spectral sums of random matrices.