Tracial joint spectral measures

Abstract

Given two Hermitian matrices, $A$ and $B$, we introduce a new type of spectral measure, a $\textit{tracial joint spectral measure}$ $\mu_{A, B}$ on the plane. Existence of this measure implies the following two results: 1) any two-dimensional subspace of the Schatten-$p$ class is isometric to a subspace of $L_{p}$, and 2) if $f : \mathbb{R} \to \mathbb{R}$ has non-negative $k$th derivative and $A$ and $B$ are Hermitian matrices with $A$ positive semidefinite, then $t \mapsto \mathrm{tr}~f(t A + B)$ has non-negative $k$th derivative. We also give an explicit expression for the measure $\mu_{A, B}$.