A Theorem on Existence of Invariant Subspaces for J-binoncontractive Operators

Let H be a J-space and let V = ((V1)(V21) (V12)(V2)) be the matrix representation of a J-binoncontractive operator V with respect to the canonical decomposition H = H+ circle plus H- of H. The main aim of this paper is to show that the assumption V-12 (V-2 - V21V1-1V12) is an element of G(infinity) implies the existence of a. V-invariant maximal nonnegative subspace. Let us note that (0.1) is a generalization of the well-known M.G. Krein condition V-12 is an element of (sic)(infinity) The set of all operators satisfying (0.1) is described via Potapov-Ginsburg transform.