On the binary relation ${\le }_u$ on self-adjoint Hilbert space operators

Abstract

Given self-adjoint operators $A,B\in \mathbb{B}(\mathcal{H})$ it is said $A{\le }_{u}B$ whenever $A\le U^{\ast }BU$ for some unitary operator $U$. We show that $A{\le }_{u}B$ if and only if $f(g(A{)}^r){\le }_{u}f(g(B{)}^r)$ 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 $B\le A\le U^{\ast }BU$, then $B=A=U^{\ast }BU$. Finally we prove that if $A^n\le U^{\ast }{A}^{n}U$ for all $n\in \mathbb{N}$, then $A=U^{\ast }AU$.

To appear in C. R. Math. Acad. Sci. Paris