Approximate Diagonal Integral Representations and Eigenmeasures for Lipschitz Operators on Banach Spaces

Ezgi Erdoğan and Enrique A. Sánchez Pérez
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Ezgi Erdoğan: Department of Mathematics, Faculty of Art and Science, University of Marmara, Kadıköy, Istanbul 34722, Turkey
Enrique A. Sánchez Pérez: Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain

Mathematics, 2022, vol. 10, issue 2, 1-24

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.

Keywords: eigenmeasure; operators; banach space; eigenvalue; spectral function (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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