The invariant subspace problem for contractive operators in Krein spaces

Abstract

The invariant subspace problem, which is one of the most fundamental questions in operator theory and which has been a subject of study for several decades remains open even on a Hilbert space. However, for certain classes of operators, for example, compact operators, solutions do exist. In our work, we address the question of invariant subspace problem for contractive operators T defined on a Krein space K. In this case we investigate the existence of semi-definite invariant subspaces. We establish that every contractive operator T defined on a Krein space K has maximal semi-definite invariant subspaces.