Continuity of the Spectrum of a Field of Self-Adjoint Operators

Abstract

Given a family of self-adjoint operators \({(A_t)_{t \in T}}\) indexed by a parameter t in some topological space T, necessary and sufficient conditions are given for the spectrum \({\sigma(A_t)}\) to be Vietoris continuous with respect to t. Equivalently the boundaries and the gap edges are continuous in t. If (T, d) is a complete metric space with metric d, these conditions are extended to guarantee Hölder continuity of the spectral boundaries and of the spectral gap edges. As a corollary, an upper bound is provided for the size of closing gaps.

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Correspondence to Siegfried Beckus.

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Work supported in part by NSF Grant DMS-1160962.

Communicated by Jan Dereziński.

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Beckus, S., Bellissard, J. Continuity of the Spectrum of a Field of Self-Adjoint Operators. Ann. Henri Poincaré 17, 3425–3442 (2016). https://doi.org/10.1007/s00023-016-0496-3

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