Extensions of Lieb's concavity theorem

TL;DR

This paper explores the convexity and concavity properties of operator functions related to Lieb's theorem, providing new proofs and extending the theorem to broader classes of functions and operator domains.

Contribution

It offers a new proof of Lieb's concavity theorem and extends the convexity and concavity results to more general operator functions involving Hilbert Schmidt operators.

Findings

Proves convexity of a class of operator functions for operator convex functions.

Provides a new proof of Lieb's concavity theorem for specific trace functions.

Establishes concavity of a new operator function involving inverse operators.

Abstract

The operator function (A,B)\to\tr f(A,B)(K^*)K, defined on pairs of bounded self-adjoint operators in the domain of a function f of two real variables, is convex for every Hilbert Schmidt operator K, if and only if f is operator convex. As a special case we obtain a new proof of Lieb's concavity theorem for the function (A,B)\to\tr A^pK^*B^{q}K, where p and q are non-negative numbers with sum p+q\le 1. In addition, we prove concavity of the operator function (A,B)\to \tr(A(A+\mu_1)^{-1}K^* B(B+\mu_2)^{-1}K) on its natural domain D_2(\mu_1,\mu_2), cf. Definition 4.1