Geometric spectral theory for compact operators

2013/09/17 by Isaak Chagouel, Chagouel, Isaak, Michael Stessin +3 · 2 citations
Mathematics · #47A10 #47A13 #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1309.4375

openalex publication_date 2013/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce a notion of joint spectrum for a tuple of compact operators on a separable Hilbert space and show that in many situations these operators commute if and only if the joint spectrum consists of countably many, locally finite, complex hyperplanes. In particular, we show that normal matrices (of the same size) A₁,⋯,A_n commute if and only if the polynomial det(z₁A₁+⋯+z_nA_n+I) is completely reducible, that is, it can be factored into a product of linear polynomials.

Cited by

• Coxeter groups and the proper joint spectrums of their faithful representations
• Spectral test of reducibility for Matrix tuples

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