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Convergence of spectral structures: a functional analytic theory and its applications to spectral geometry. (English) Zbl 1092.53026

From the authors’ abstract: We present a functional analytic framework of some natural topologies on a given family of spectral structures on Hilbert spaces, and study convergence of Riemannian manifolds and their spectral structure induced from the Laplacian. We also consider convergence of Alexandrov spaces, locally finite graphs, and metric spaces with Dirichlet forms. Our study covers convergence of noncompact (or incomplete) spaces whose Laplacian has continuous spectrum.

MSC:

53C20 Global Riemannian geometry, including pinching
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
31C25 Dirichlet forms
46C99 Inner product spaces and their generalizations, Hilbert spaces
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