Antitonicity of the Inverse and J-Contractivity

Abstract

A well-known result in operator theory states that the inverse operator function is antitone, relative to the usual ordering by positivity, on the set of positive invertible operators, i.e., given two such operators A and B, (⋆)A ≤ B ⇒ A −1 ≥ B −1. In this paper recent work by Shmul’yan [S] and the authors [HN] on the extension of (⋆) to the case of selfadjoint invertible A and B is reviewed. Some of the results in [S] and [HN] are here given different proofs. In particular, a criterion for (⋆) in terms of the spectrum of B −1 A is proved without relying on results from the theory of indefinite inner product spaces. The geometrical aspects of (⋆) are explored, and are shown to lead naturally to the study of convex sets of invertible selfadjoint operators. In particular, criteria for a set of operators to be coverable by a convex set of invertible selfadjoint operators are derived. These results are used to obtain additional (geometrical) characterizations of (⋆) as well as further criteria for the J-bicontractivity of J-contractions in a Kreĭn space.