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

Given a family of self-adjoint operators ( A t ) t ∈ T indexed by a parameter t in some topological space T , necessary and sufficient conditions are given for the spectrum σ ( 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.