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Transfer Operators on L1 and L2
Book chapter   Peer reviewed

Transfer Operators on L1 and L2

Sergey Bezuglyi and Palle E. T Jorgensen
Transfer Operators, Endomorphisms, and Measurable Partitions, pp.59-76
Lecture Notes in Mathematics, Springer International Publishing
06/22/2018
DOI: 10.1007/978-3-319-92417-5_5

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Abstract

Given a transfer operator (R, σ), it is of interest to find the measuresμ such that both R and σ induce operators in the corresponding Lp spaces, i.e., in Lp(X,ℬ,μ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$L^p(X, {\mathcal B}, \mu )$$ \end{document}. We turn to this below, but our main concern are the cases p = 1, p = 2, and p = ∞. When R is realized as an operator in L2(X,ℬ,μ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$L^2(X, {\mathcal B}, \mu )$$ \end{document}, for a suitable choice of μ, then it is natural to ask for the adjoint operator R∗ where “adjoint” is defined with respect to the L2(μ)-inner product.
Induce operators Infinite-dimensional Perron-Frobenius theorem Universal Hilbert space

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