The Functional Calculus Approach to the Spectral Theorem
Functional Analysis
2020-09-28 v2 Operator Algebras
Spectral Theory
Abstract
A consistent functional calculus approach to the spectral theorem for strongly commuting normal operators on Hilbert spaces is presented. In contrast to the common approaches using projection-valued measures or multiplication operators, here the functional calculus is not treated as a subordinate but as the central concept. Based on five simple axioms for a "measurable functional calculus", the theory of such calculi is developed in detail, including spectral theory, uniqueness results and construction principles. Finally, the functional calculus form of the spectral theorem is stated and proved, with some proof variants being discussed.
Cite
@article{arxiv.2003.06130,
title = {The Functional Calculus Approach to the Spectral Theorem},
author = {Markus Haase},
journal= {arXiv preprint arXiv:2003.06130},
year = {2020}
}
Comments
38 pages. Slightly revised. Accepted for publication in Indagationes Mathematicae