Spectral decomposition of self-adjoint operators in Pontryagin and Krein spaces

V. A. Shtraus

South Ural State University (national research university), Chelyabinsk, Russia

Abstract: We consider a self-adjoint operator acting in a Krein space and possessing an invariant subspace that is maximal nonnegative and decomposes into a direct sum of a uniformly positive (i.e., equivalent to a Hilbert space with respect to the inner pseudoscalar product) and a finite-dimensional neutral subspace. We prove the existence of a difference expression that transforms the moment sequence generated by this operator into a sequence representable as the difference of positive moment sequences. In the case of a cyclic operator, this result is applied to construct a function space in which the operator under study is modeled as the operator of multiplication by an independent variable.

Keywords: self-adjoint operator, Krein space, Pontryagin space, invariant subspace, spectral decomposition.

UDC: 517.982+517.984.48

DOI: 10.22363/2413-3639-2025-71-3-524-546