Further developments of Furuta inequality of indefinite type

Abstract

A selfadjoint involutive matrix J endows Cn with an indefinite inner product [•,•] given by [x,y] := (Jx,y), x,y ∈ C n. We study matrix inequalities for J-selfadjoint matrices with nonnegative eigenvalues. Namely, Furuta inequality of indefinite type is revisited. Characterizations of the J-chaotic order and of the J-relative entropy are obtained via Furuta inequality. The parallelism between the definite versions of the inequalities on Hilbert spaces and the corresponding indefinite versions on Krein spaces is pointed out.