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 (H , J) is a Krein space, U is an open set which is symmetric with respect to the real axis such that U ∩ R consists of a segment of real axis and f is a Kreinoperator convex function on U with f(0) = 0, then f(CAC) ≤ Cf(A)C for all J-positive operators A and all invertible J-contractions C such that the spectra of A, CAC and DAD are contained in U , where D is a defect operator for C. We also show that in contrast with usual operator convex functions the converse of this implication is not true, in general.