On The Continuity of the Spectrum of Fields of Operators

Abstract

In their article, "Continuity of the Spectrum of a Field of Self-Adjoint Operators", Beckus and Bellissard discuss the following problem: Given a family of bounded, self-adjoint linear operators indexed by a parameter t in some topological space T, what are the different interesting forms of continuity of the spectrum of these operators, and how are the different types of continuity connected? They conclude that for bounded, self-adjoint operators, there is an equivalence between continuity of the gap edges, p2-continuity of the field and Fell continuity of the spectrum; furthermore, for metric space settings, they give results relating p2-α-Hölder continuity of the field with α-Hölder continuity of the gap edges and α/2-Hölder continuity of the width of gaps. In this article, we will extend several of Beckus and Bellissard’s ideas, as well as giving detailed and precise proofs of all their claims. This is particularly true in the case of unbounded, self-adjoint operators, which are only treated very briefly by Beckus and Bellisard.