2015/09/21 by Michael Stessin, Stessin, Michael, Alexandre Tchernev +1
Mathematics · #14J70 #14P15 #47A20 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #FOS: Mathematics #Holomorphic and Operator Theory #Primary: 47A25 #Secondary: 47A80 #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1509.06274
openalex publication_date 2015/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Joint spectra of tuples of operators are subsets in complex projective space. The corresponding tuple of operators can be viewed as an infinite dimensional analog of a determinantal representation of the joint spectrum. We investigate the relationship between the geometry of the spectrum and the properties of the operators in the tuple when these operators are self-adjoint. In the case when the spectrum contains an algebraic curve passing through an isolated spectral point of one of the operators we give necessary and sufficient geometric conditions for the operators in the tuple to have a common reducing subspace. We also address spectral continuity and obtain a norm estimate for the commutant of a pair of self-adjoint matrices in terms of the Hausdorff distance of their joint spectrum to a family of lines.