The Functional Calculus Approach to the Spectral Theorem

2020/03/13 by Haase, Markus
#FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Primary 47B15 #Secondary 46A60 #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2003.06130

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.

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