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.
Categories: math.FA math.OA