Note on the Spectral Theorem

TL;DR

This paper discusses two representations of the spectral theorem, introducing a new 'deformed' representation that extends to broader classes of operators and spaces, enhancing the theorem's applicability.

Contribution

It introduces the deformed representation of the spectral theorem, extending its applicability to all closed densely defined operators on Hilbert spaces and beyond.

Findings

Deformed representation simplifies extension to all closed densely defined operators.

It extends to all separable reflexive Banach spaces.

Limited extension to non-reflexive Banach spaces.

Abstract

In this note, we show that the spectral theorem, has two representations; the Stone-von Neumann representation and one based on the polar decomposition of linear operators, which we call the deformed representation. The deformed representation has the advantage that it provides an easy extension to all closed densely defined linear operators on Hilbert space. Furthermore, the deformed representation can also be extended all separable reflexive Banach spaces and has a limited extension to non-reflexive Banach spaces.