Basic representation theorems of forms

TL;DR

This paper explores the representation of nonnegative sesquilinear forms in Hilbert spaces, establishing connections with positive self-adjoint operators and providing insights into Friedrichs extensions.

Contribution

It introduces new representation theorems for non-closed nonnegative forms and relates them to positive self-adjoint operators, extending classical results.

Findings

Established a correspondence between nonnegative forms and self-adjoint operators.

Provided a simplified proof of Friedrichs extension for positive operators.

Extended classical representation theorems to non-closable forms.

Abstract

We study maximal representations of nonnegative sesquilinear forms in real or complex Hilbert spaces, that are not necessarily closed or even closable. We associate positive self-adjoint operators with such forms, in a sense similar to Kato's representation theorems. In particular, we give a brief proof of the Friedrichs extension of a densely defined positive operator.