Operator convexity in Krein spaces

TL;DR

This paper extends the concept of operator convexity to Krein spaces, establishing an indefinite Jensen inequality and exploring the properties and limitations of Krein-operator convex functions.

Contribution

It introduces Krein-operator convexity, formulates an indefinite Jensen inequality in Krein spaces, and analyzes the differences from classical operator convex functions.

Findings

Established an indefinite Jensen operator inequality for Krein spaces.

Showed that the converse of the inequality does not hold generally.

Highlighted differences between Krein-operator convex functions and classical operator convex functions.

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.