Operator maps of Jensen-type

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 \(\Phi :\mathbb {B}_J({\mathcal {H}})\rightarrow {{\mathbb {B}}}({\mathcal {H}})_\text {sa} \) is said to be of Jensen-type if

$$\begin{aligned} \Phi (C^*AC+D^*BD)\le C^*\Phi (A)C+D^*\Phi (B)D \end{aligned}$$

for all \( A, B \in \mathbb {B}_J({\mathcal {H}})\) 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 an infinite dimensional Hilbert space is of the form \(\Phi (A)=f(A)\) for some operator convex function f defined in J.