CmPositive Eigenvectors for Linear Operators Arising in the Computation of Hausdorff Dimension

Roger D. Nussbaum

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1 Scopus citations

Abstract

We consider a broad class of linear Perron–Frobenius operators (Formula presented.) , where (Formula presented.) is a real Banach space of (Formula presented.) functions. We prove the existence of a strictly positive (Formula presented.) eigenvector (Formula presented.) with eigenvalue (Formula presented.) the spectral radius of (Formula presented.). We prove (see Theorem 6.5 in Sect. 6 of this paper) that (Formula presented.) is an algebraically simple eigenvalue and that, if (Formula presented.) denotes the spectrum of the complexification of (Formula presented.) , where (Formula presented.). Furthermore, if (Formula presented.) is any strictly positive function, (Formula presented.) as (Formula presented.) , where (Formula presented.) and convergence is in the norm topology on (Formula presented.). In applications to the computation of Hausdorff dimension, one is given a parametrized family (Formula presented.) , of such operators and one wants to determine the (unique) value (Formula presented.) such that (Formula presented.). In another paper (Falk and Nussbaum in C (Formula presented.) Eigenfunctions of Perron–Frobenius operators and a new approach to numerical computation of Hausdorff dimension, submitted) we prove that explicit estimates on the partial derivatives of the positive eigenvector (Formula presented.) of (Formula presented.) can be obtained and that this information can be used to give rigorous, sharp upper and lower bounds for (Formula presented.).

Original languageEnglish (US)
Pages (from-to)357-393
Number of pages37
JournalIntegral Equations and Operator Theory
Volume84
Issue number3
DOIs
StatePublished - Mar 1 2016

All Science Journal Classification (ASJC) codes

  • Analysis
  • Algebra and Number Theory

Keywords

  • C positive eigenvector
  • Essential spectrum
  • Generalized Krein–Rutman theorem
  • Measures of noncompactness
  • Radius of essential spectrum

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