Operators on positive semidefinite inner product spaces

Abstract

Let U be a semiunitary space; i.e., a complex vector space with scalar product given by a positive semidefinite Hermitian form 〈⋅,⋅〉. If a linear operator A:U→U is bounded (i.e., ‖Au‖⩽c‖u‖ for some c∈R and all u∈U), then the subspace U0:={u∈U|〈u,u〉=0} is invariant, and so A defines the linear operators A0:U0→U0 and A1:U/U0→U/U0.Let A be an indecomposable bounded operator on U such that 0≠U0≠U. Let λ be an eigenvalue of A0. We prove that the algebraic multiplicity of λ in A1 is not less than the geometric multiplicity of λ in A0, and the geometric multiplicity of λ in A1 is not less than the number of Jordan blocks Jt(λ) of each fixed size t×t in the Jordan canonical form of A0.We give canonical forms of selfadjoint and isometric operators on U, and of Hermitian forms on U.For an arbitrary system of semiunitary spaces and linear mappings on/between them, we give an algorithm that reduces their matrices to canonical form. Its special cases are Belitskii's and Littlewood's algorithms for systems of linear operators on vector spaces and unitary spaces, respectively.