Tracial joint spectral measures

Abstract

Given two Hermitian matrices, $A$ and $B$, we introduce a new type of spectral measure, a $\text{tracial joint spectral measure}$ ${\mu }_{A,B}$ 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 $t\mapsto \mathrm{tr}f(tA+B)$ has non-negative $k$th derivative. We also give an explicit expression for the measure ${\mu }_{A,B}$.

15 pages, 1 figure