Operator inequalities for J-contractions

Abstract

A selfadjoint involutive matrix J endows Cn with an indefinite inner product [·, ·] given by [x,y] = ≤Jx,y , x,y ∈ Cn. Characterizations of the J -chaotic order Log(A) ≤J Log(B) are presented for J -selfadjoint matrices A,B with positive eigenvalues, in terms of operator functions involving the α -power mean and the J -relative entropy. An indefinite complete form of the Furuta inequality and some exponential operator inequalities for J -selfadjoint matrices are also obtained. The parallelism between the inequalities in Hilbert spaces and the corresponding indefinite versions in Krein spaces is pointed out.