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Positivity

, Volume 22, Issue 5, pp 1255–1263 | Cite as

Operator maps of Jensen-type

  • Frank Hansen
  • Mohammad Sal Moslehian
  • Hamed Najafi
Article

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 \(\Phi :\mathbb {B}_J({\mathcal {H}})\rightarrow {{\mathbb {B}}}({\mathcal {H}})_\text {sa} \) 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 CD 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.

Keywords

Jensen’s operator inequality Convex operator function 

Mathematics Subject Classification

Primary 47A63 Secondary 47B10 47A30 

Notes

Acknowledgements

The third author was supported by a grant from Ferdowsi University of Mashhad (No. 2/46186).

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute for Excellence in Higher EducationTohoku UniversitySendaiJapan
  2. 2.Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS)Ferdowsi University of MashhadMashhadIran

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