Operator convexity in Krein spaces

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.

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