Operator maps of Jensen-type

Abstract

Let ${\mathbb{B}}_J(\mathcal{H})$ denote the set of self-adjoint operators acting on a Hilbert space $\mathcal{H}$ with spectra contained in an open interval $J$. A map $\Phi :{\mathbb{B}}_J(\mathcal{H})\to \mathbb{B}(\mathcal{H}{)}_{\text{sa}}$ is said to be of Jensen-type if \[ Φ(C^*AC+D^*BD)\le C^*Φ(A)C+D^*Φ(B)D \] for all $A,B\in B_J(\mathcal{H})$ and bounded linear operators $C,D$ acting on $\mathcal{H}$ with $C^{\ast }C+D^{\ast }D=I$, where $I$ denotes the identity operator. We show that a Jensen-type map on a infinite dimensional Hilbert space is of the form $\Phi (A)=f(A)$ for some operator convex function $f$ defined in $J$.

8 pages