## 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 U_{0}:={u∈U|〈u,u〉=0} is invariant, and so A defines the linear operators A_{0}:U_{0}→U_{0} and A_{1}:U/U_{0}→U/U_{0}. Let A be an indecomposable bounded operator on U such that 0≠U_{0}≠U. Let λ be an eigenvalue of A_{0}. We prove that the algebraic multiplicity of λ in A_{1} is not less than the geometric multiplicity of λ in A_{0}, and the geometric multiplicity of λ in A_{1} is not less than the number of Jordan blocks J_{t}(λ) of each fixed size t×t in the Jordan canonical form of A_{0}. 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.

Original language | English |
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Pages (from-to) | 82-105 |

Number of pages | 24 |

Journal | Linear Algebra and Its Applications |

Volume | 596 |

DOIs | |

Publication status | Published - Jul 1 2020 |

## Keywords

- Belitskii's and Littlewood's algorithms
- Bounded operators
- Positive semidefinite inner product spaces
- Selfadjoint and isometric operators

## ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics