Representation Theorems for indefinite quadratic forms without spectral gap

TL;DR

This paper extends the representation theorems for indefinite quadratic forms to include unbounded cases without spectral gaps, broadening the understanding of associated operators and their kernels.

Contribution

It generalizes existing theorems to unbounded forms without spectral gaps, including new cases and kernel characterizations.

Findings

Extended the First and Second Representation Theorems for indefinite quadratic forms.

Included new unbounded cases of operators without spectral gaps.

Determined kernels of associated operators in special cases.

Abstract

The First and Second Representation Theorem for sign-indefinite quadratic forms are extended. We include new cases of unbounded forms associated with operators that do not necessarily have a spectral gap around zero. The kernel of the associated operators is determined for special cases. This extends results by Grubi\v{s}i\'c, Kostrykin, Makarov and Veseli\'c in [Mathematika 59 (2013), 169--189].