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

Abstract

Let $\mathcal{H}$ be a J-space and let $V = \left( {\begin{array}{*{20}c} {V_1 } & {V_{12} } \\ {V_{21} } & {V_2 } \\ \end{array} } \right)$ be the matrix representation of a J-binoncontractive operator V with respect to the canonical decomposition $\mathcal{H} = \mathcal{H}^ + \oplus \mathcal{H}^ -$ of $\mathcal{H}$. The main aim of this paper is to show that the assumption

$$ V_{12} (V_2 - V_{21} V_1^{ - 1} V_{12} ) \in \mathfrak{S}_\infty $$

((0.1))

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 $V_{12} \in \mathfrak{S}_\infty$. The set of all operators satisfying (0.1) is described via Potapov-Ginsburg transform.

The research of T.Ya. Azizov was supported by the grant of RFBR 08-01-00566-a.