Self-Adjoint Extension of Symmetric Maps

TL;DR

This paper discusses the extension of densely-defined symmetric linear maps in real Hilbert spaces to self-adjoint maps, highlighting conditions for uniqueness and explicit extension methods.

Contribution

It provides a new expression for self-adjoint extension using Riesz representation and characterizes when the extension is unique, including the Friedrichs extension case.

Findings

Self-adjoint extension can be expressed via Riesz representation.

Uniqueness of extension occurs in the Friedrichs extension case.

Extension equals the closure of the original map in certain cases.

Abstract

A densely-defined symmetric linear map from/to a real Hilbert space extends to a self-adjoint map. Extension is expressed via Riesz representation. For a case including Friedrichs extension of a strongly monotone map, self-adjoint extension is unique, and equals closure of the given map.