Tracial joint spectral measures

Abstract

Trace inequalities, that is inequalities between traces of complex matrices, are ubiquitous in various branches of mathematics. While such inequalities are usually easy to state as generalizations of real variable inequalities, proving them often requires deep understanding. We introduce a new general tool for investigating trace inequalities, namely the tracial joint spectral measure. This positive measure on the plane can be associated to any two Hermitian matrices, and existence of it implies a plethora of non-trivial trace inequalities for these matrices. In chapter $1$, we discuss existence and basic properties of these measures, giving an explicit expression for them along the way. As the first main application, we deduce a new tracial monotonicity property: if $f$ has non-negative $k$:th derivative, then so does $t \mapsto \tr f(t A + B)$ for any Hermitian $A, B$ with $A$ positive definite. In chapter $2$, we apply the theory of tracial joint spectral measures to Schatten-$p$ trace ideals. In this context, we give a new embedding result: any two-dimensional subspace of Schatten-$p$ is isometric to a subspace of $L_{p}$. This result is used to resolve a conjecture of Ball, Carlen, and Lieb on the extension of Hanner's inequality to Schatten-$p$ spaces. Finally, we discuss the ways in which our embedding result fails for more than two matrices/operators and investigate ideas for working in this higher dimensional setting.