New York Journal of Mathematics, ( ISI ), Volume (20), No (1), Year (2014-1) , Pages (133-144)

#### Title : ( Operator convexity in Krein spaces )

Authors: Mohammad Sal Moslehian , Mahdi Dehghani ,

Citation: BibTeX | EndNote

#### Abstract

We introduce the notion of Krein-operator convexity in the setting of Krein spaces. We present an indefinite version of the Jensen operator inequality on Krein spaces by showing that if $(\mathscr{H},J)$ is a Krein space, $\mathcal{U}$ is an open set which is symmetric with respect to the real axis such that $\mathcal{U}\cap\mathbb{R}$ consists of a segment of real axis and $f$ is a Krein-operator convex function on $\mathcal{U}$ with $f(0)=0$, then \begin{eqnarray*} f(C^{\sharp}AC)\leq^{J}C^{\sharp}f(A)C \end{eqnarray*} for all $J$-positive operators $A$ and all invertible $J$-contractions $C$ such that the spectra of $A$, $C^{\sharp}AC$ and $D^{\sharp}AD$ are contained in $\mathcal{U}$, where $D$ is a defect operator for $C^{\sharp}$.\\ We also show that in contrast with usual operator convex functions the converse of this implication is not true, in general.

#### Keywords

, Indefinite inner product;J, contraction;J , selfadjoint operator; Julia operator; Krein space; Krein, operator convex function
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@article{paperid:1041237,
author = {Sal Moslehian, Mohammad and Dehghani, Mahdi},
title = {Operator convexity in Krein spaces},
journal = {New York Journal of Mathematics},
year = {2014},
volume = {20},
number = {1},
month = {January},
issn = {1076-9803},
pages = {133--144},
numpages = {11},
keywords = {Indefinite inner product;J-contraction;J -selfadjoint operator; Julia operator; Krein space; Krein-operator convex function},
}