On the binary relation $\leq_u$ on self-adjoint Hilbert space operators

TL;DR

This paper characterizes a unitary-based order relation on self-adjoint operators using operator convex and monotone functions, providing conditions for equality and invariance under unitary conjugation.

Contribution

It establishes a new characterization of the relation $A rianglelefteq_u B$ via functional calculus and proves conditions under which operators are unitarily equivalent.

Findings

$A rianglelefteq_u B$ iff $f(g(A)^r) rianglelefteq_u f(g(B)^r)$ for specified functions

Conditions under which $B rianglelefteq A rianglelefteq_u B$ imply $A=B$

If $A^n rianglelefteq U^*A^nU$ for all $n$, then $A$ is unitarily invariant

Abstract

Given self-adjoint operators $A, B\in\mathbb{B}(\mathscr{H})$ it is said $A\leq_uB$ whenever $A\leq U^*BU$ for some unitary operator $U$. We show that $A\leq_u B$ if and only if $f(g(A)^r)\leq_uf(g(B)^r)$ for any increasing operator convex function $f$, any operator monotone function $g$ and any positive number $r$. We present some sufficient conditions under which if $B\leq A\leq U^*BU$, then $B=A=U^*BU$. Finally we prove that if $A^n\leq U^\ast A^nU$ for all $n\in\mathbb{N}$, then $A=U^\ast AU$.