Completion and extension of operators in Kre\u{\i}n spaces

TL;DR

This paper extends Kre
2bf1's classical results on selfadjoint contractive extensions to operators in Kre
2bf1 and Pontryagin spaces, allowing for a fixed number of negative eigenvalues, thus broadening the scope of operator extension theory.

Contribution

It generalizes Kre
2bf1's extension results and Shmul'yan's completion theorem to operators with a fixed number of negative eigenvalues in Kre
2bf1 spaces.

Findings

Extended the description of selfadjoint contractive extensions in Kre
2bf1 spaces.

Generalized Shmul'yan's completion results for operators with negative eigenvalues.

Provided new theoretical tools for operator extension in indefinite inner product spaces.

Abstract

A generalization of the well-known results of M.G. Kre\u{\i}n about the description of selfadjoint contractive extension of a hermitian contraction is obtained. This generalization concerns the situation, where the selfadjoint operator $A$ and extensions $\widetilde A$ belong to a Kre\u{\i}n space or a Pontryagin space and their defect operators are allowed to have a fixed number of negative eigenvalues. Also a result of Yu.L. Shmul'yan on completions of nonnegative block operators is generalized for block operators with a fixed number of negative eigenvalues in a Kre\u{\i}n space. This paper is a natural continuation of S. Hassi's and author's paper [5].