A Reducing of the Invariant Semidefinite Subspace Problem for Krein Noncontraction to such a Problem for Krein Isometry

Abstract

Definition. Let J be a period-2 unitary operator (some people say J is reflection operator or reflection symmetry) and U be a linear operator. If U^*JU = J (resp. U^*JU >= J) then U is said to be J-isometry (resp. J-noncontraction). If U^*JU >= J and UJU^* >= J) then U is said to be J-binoncontraction). Theorem. If every J-isometry has nontrivial positive invariant subspace then every J-noncontraction has such a subspace. Theorem. If every J-binoncontractive J-isometry has maximal positive invariant subspace then every J-noncontraction has such a subspace. The article text is the complete text of the author's report on 15-th Voronezh Winter Mathematical School, p 119 (see. VINITI 16.12.81, N 5691-81). But in that time the presented construtions and theorems seemed to be rather curious observations. Now the situattion is changing (see e.g. math.DS/9908169)