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.