Abstract
We introduce the notion of Kreinoperator 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(C^{#}AC)≦^{J}C^{#}f(A)C
for all Jpositive operators A and all invertible Jcontractions C
such that the spectra of
A, C^{#}AC and D^{#}AD 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.

Author information
Mohammad Sal Moslehian:
Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91775, Iran
moslehian@member.ams.org
Mahdi Dehghani:
Department of Pure and Applied Mathematics, University of Yazd, Yazd 89195741, Iran
Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91775, Iran
e.g.mahdi@gmail.com
