Selfadjoint operators in S-spaces
We study S-spaces and operators therein. An S-space is a Hilbert space with an additional inner product given by, where U is a unitary operator in. 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 with $ρ(A) ̸= ∅$ we find an inner product which turns S into a Krein space and A into a selfadjoint operator therein. In addition, we give a simple condition for the existence of invariant subspaces.
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