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 H\"older 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.