Operator maps of Jensen-type
Frank Hansen, Mohammad Sal Moslehian, and Hamed Najafi

TL;DR
This paper characterizes Jensen-type maps on self-adjoint operators in infinite-dimensional Hilbert spaces, showing they are precisely operator functions derived from operator convex functions.
Contribution
It provides a complete characterization of Jensen-type maps as operator functions from operator convex functions on the spectrum interval.
Findings
Jensen-type maps are of the form (A) for some operator convex function f.
The characterization holds in infinite-dimensional Hilbert spaces.
The result extends the understanding of operator inequalities and functional calculus.
Abstract
Let denote the set of self-adjoint operators acting on a Hilbert space with spectra contained in an open interval . A map is said to be of Jensen-type if \[ \Phi(C^*AC+D^*BD)\le C^*\Phi(A)C+D^*\Phi(B)D \] for all and bounded linear operators acting on with , where denotes the identity operator. We show that a Jensen-type map on a infinite dimensional Hilbert space is of the form for some operator convex function defined in .
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