Abstract
A new stochastic approach for the approximation of (nonlinear) Lipschitz operators in normed spaces by their eigenvectors is shown. Different ways of providing integral representations for these approximations are proposed, depending on the properties of the operators themselves whether they are locally constant, (almost) linear, or convex. We use the recently introduced notion of eigenmeasure and focus attention on procedures for extending a function for which the eigenvectors are known, to the whole space. We provide information on natural error bounds, thus giving some tools to measure to what extent the map can be considered diagonal with few errors. In particular, we show an approximate spectral theorem for Lipschitz operators that verify certain convexity properties.
Highlights
Diagonalization of operators is deeply tied to the linear character of the operators themselves
A fruitful diagonalization theory has to provide a way to compute the eigenvectors, as well as a structural rule to extend the representation of the map from the eigenvectors to the whole space. This is the point of view from which we develop the research presented in this paper: we consider the set of eigenvalues of a given map and use a given rule to approximate the original map starting from them
Much research has been done for finding results on the spectrum of a Lipschitz operator that could be similar to those that hold in the linear case [1]
Summary
Diagonalization of operators is deeply tied to the linear character of the operators themselves. A fruitful diagonalization theory has to provide a way to compute the eigenvectors, as well as a structural rule to extend the representation of the map (or at least an approximation) from the eigenvectors to the whole space This is the point of view from which we develop the research presented in this paper: we consider the set of eigenvalues of a given map and use a given rule to approximate the original map starting from them. We intend to show to what extent this tool can be used to obtain quasi-diagonal representations of T for the case of linear operators We will tackle this question in the following subsection for the canonical case, i.e., the Euclidean case in which v is given by the average of Dirac’s deltas of the elements of an orthonormal basis
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