Note on the Spectral Theorem

2012/10/31 by Tepper L. Gill, Daniel R. Williams, Gill, Tepper L +1
Computer Science · Mathematics · #35Q80 #46B03 #47D03 #47F05 #47H06 #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1211.0058

openalex publication_date 2012/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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