On the binary relation ≤u on self-adjoint Hilbert space operators

Given self-adjoint operators A, B is an element of B(H) it is said A <=(u) B whenever A <= U*BU for some unitary operator U. We show that A <=(u) B if and only if f(g(A)(r)) <=(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 <= A <= U*BU, then B = A = U*BU. Finally we prove that if A(n) <= U*A(n)U for all n is an element of N, then A = U*AU. (C) 2012 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.