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\colon\mathbb{B}_J(\mathcal H)\to {\mathbb B}(\mathcal H)_\text{sa} $ is said to be of Jensen-type if \[ \Phi(C^*AC+D^*BD)\le C^*\Phi(A)C+D^*\Phi(B)D \] for all $ A, B \in B_J(\mathcal H)$ and bounded linear operators $ C,D $ acting on $ \mathcal H $ with $ C^*C+D^*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 $.