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Operator maps of Jensen-type
- Source :
- Positivity 22 (2018), no. 5, 1255-1263
- Publication Year :
- 2017
-
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 $.<br />Comment: 8 pages
Details
- Database :
- arXiv
- Journal :
- Positivity 22 (2018), no. 5, 1255-1263
- Publication Type :
- Report
- Accession number :
- edsarx.1708.07028
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1007/s11117-018-0571-8