Selfadjoint operators in S-spaces

Abstract

We study S-spaces and operators therein. An S-space is a Hilbert space ( S , ( ⋅ , − ) ) with an additional inner product given by [ ⋅ , − ] : = ( U ⋅ , − ) , where U is a unitary operator in ( S , ( ⋅ , − ) ) . We investigate spectral properties of selfadjoint operators in S-spaces. We show that their spectrum is symmetric with respect to the real axis. As a main result we prove that for each selfadjoint operator A in an S-space we find an inner product which turns S into a Krein space and A into a selfadjoint operator therein. As a consequence we get a new simple condition for the existence of invariant subspaces of selfadjoint operators in Krein spaces, which provides a different insight into this well-know and in general unsolved problem.