Various form closures associated with a fixed non-semibounded self-adjoint operator

TL;DR

This paper explores the various closures of sesquilinear forms associated with a fixed non-semibounded self-adjoint operator, revealing their structure, spectral properties, and the potential for more information than the operators themselves.

Contribution

It establishes a correspondence between closed symmetric forms with gap point 0 and Krein space operators, and analyzes their spectral functions, especially near infinity.

Findings

Multiple closures of a form can exist for non-semibounded operators.

A one-to-one correspondence is found between certain forms and Krein space operators.

Spectral analysis near infinity reveals detailed properties of these closures.

Abstract

If $T$ is a semibounded self-adjoint operator in a Hilbert space $(H, \, (\cdot , \cdot))$ then the closure of the sesquilinear form $(T \cdot , \cdot)$ is a unique Hilbert space completion. In the non-semibounded case a closure is a Kre\u{\i}n space completion and generally, it is not unique. Here, all such closures are studied. A one-to-one correspondence between all closed symmetric forms (with ``gap point'' $0$) and all J-non-negative, J-self-adjoint and boundedly invertible Kre\u{\i}n space operators is observed. Their eigenspectral functions are investigated, in particular near the critical point infinity. An example for infinitely many closures of a fixed form $(T \cdot , \cdot)$ is discussed in detail using a non-semibounded self-adjoint multiplication operator $T$ in a model Hilbert space. These observations indicate that closed symmetric forms may carry more information than self-adjoint Hilbert space operators.