Operator Theory and Analysis of Infinite Networks

This volume presents resistance networks that study large graphs which are connected, undirected, and weighted. Such networks provide a discrete model for diffusion and other physical processes in inhomogeneous media, and arise in a host of applications, including the flow of heat in perforated media, and diffusion of water in porous matter. Variants of these graph models are also of direct relevance to data science, e.g., the case of transfer of data through the internet, and the propagation of memes through social networks. Indeed, network analysis plays a crucial role in many other areas of data science and engineering, e.g., in the study of connected convolutional neural networks.In keeping with the metaphor of particular networks, recall systems arising from electricity flowing through a graph-network of conductors, then the weights assigned to networks correspond to conductances (the conductance is the reciprocal of resistance), and functions on the vertices may be interpreted as voltages. The corresponding functions on the edges of the graph may then be interpreted as currents.Resistance networks also arise in probability, as they correspond to a class of Markov chains under suitable conditions. As such, the wider range of applications include random walk models, especially reversible Markov processes; and statistical inference in machine learning models. However, the present volume considers the analysis of resistance networks from the point of view of Hilbert space theory, using Dirchlet energy as an inner product. The resulting viewpoint emphasizes orthogonality over convexity and provides new insights into the connections between harmonic functions, operators, and boundary theory. Novel applications to mathematical physics, in particular quantum information, are given, especially in regard to the part of operator theory dealing with the self-adjointness question for unbounded operators.New topics are covered in a host of applied areas that is accessible to multiple audiences, including students; at both beginning and more advanced levels. This is accomplished by the authors directly linking diverse applied questions to such key areas of mathematics as functional analysis, operator theory, harmonic analysis, optimization, approximation theory, and probability theory.