Almost periodic Jacobi matrices with homogeneous spectrum, infinite dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions

Mikhail Sodin, Peter Yuditskii

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93 Scopus citations

Abstract

All three subjects reflected in the title are closely intertwined in the paper. Let JE be a class of Jacobi matrices acting in l2(ℤ) with a homogeneous spectrum E (see Definition 3.2) and with diagonal elements of the resolvent R(m, m; z) having pure imaginary boundary values a.e. on E. For this class, we extend fundamental results pertaining to the finite-band (i.e., algebraic-geometrical) operators. In particular, we prove that matrices of the class JE are almost periodic. Our main tool is a theory of character-automorphic functions with respect to the Fuchsian group uniformizing the resolvent domain. For Widom type groups we find a natural analog of the Fourier basis and for Widom-Carleson type groups we characterize the orthogonal complement to character-automorphic functions from the Hardy space H2. This technique allows us to study the infinite dimensional Abel map and to find an infinite dimensional real version of the Jacobi inversion, which play a principal role in our investigation of matrices of the class JE.

Original languageEnglish
Pages (from-to)387-435
Number of pages49
JournalJournal of Geometric Analysis
Volume7
Issue number3
DOIs
StatePublished - 1997
Externally publishedYes

Keywords

  • Almost periodic Jacobi matrices
  • Character-automorphic functions
  • Generalized Abel map
  • Jacobi inversion
  • Widom type Fuchsian groups

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