Spectral algebraic curves and decomposable operator tuples

TL;DR

This paper explores the geometric properties of joint spectra of self-adjoint operator tuples, linking algebraic curves in the spectrum to the operators' reducibility and providing spectral continuity estimates.

Contribution

It establishes geometric conditions for common reducing subspaces based on spectral algebraic curves and analyzes spectral continuity for self-adjoint matrices.

Findings

Necessary and sufficient conditions for operators to have a common reducing subspace.

Norm estimates for the commutant based on spectral Hausdorff distance.

Relationship between spectral algebraic curves and operator properties.

Abstract

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.