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

Given a family of self-adjoint operators (At)t∈T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(A_t)_{t \in T}}$$\end{document} indexed by a parameter t in some topological space T, necessary and sufficient conditions are given for the spectrum σ(At)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sigma(A_t)}$$\end{document} 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.