Tracial joint spectral measures

TL;DR

This paper introduces the tracial joint spectral measure for Hermitian matrices, establishing its existence and implications for isometric embeddings in Schatten-$p$ spaces and properties of trace functions.

Contribution

It defines a new spectral measure for pairs of Hermitian matrices and proves its existence, leading to new isometric and convexity results in matrix analysis.

Findings

Existence of the tracial joint spectral measure for Hermitian matrices.

Implication that two-dimensional Schatten-$p$ subspaces are isometric to $L_{p}$.

Trace functions with non-negative derivatives preserve convexity properties.

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 $\mu_{A, B}$.