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Frames and factorization of graph Laplacians. (English) Zbl 1359.47055

This paper provides an investigation on the existence of a Parseval frame in an energy Hilbert space and on some of its essential applications. In the first two sections, some basics are introduced. The third section of the paper is devoted to considerations on electrical current as frame coefficients and the Parseval frame is defined. The authors identify a canonical Parseval frame in the considered Hilbert space and prove that it is not an orthonormal basis (ONB) except in simple degenerate cases. By means of this frame, some explicit results are obtained. For instance, the authors investigate the Friedrichs extension of the graph Laplacian and the way in which the Parseval frame and related closable operators can be used to give a factorization of the Friedrichs extension of the Laplacian. A large number of practical examples are presented in the final section.

MSC:

47L60 Algebras of unbounded operators; partial algebras of operators
46N30 Applications of functional analysis in probability theory and statistics
46N50 Applications of functional analysis in quantum physics
42C15 General harmonic expansions, frames
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C75 Structural characterization of families of graphs
31C20 Discrete potential theory
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
58J65 Diffusion processes and stochastic analysis on manifolds
81S25 Quantum stochastic calculus
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