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

Abstract

All three subjects reflected in the title are closely intertwined in the paper. LetJ E be a class of Jacobi matrices acting inl 2(ℤ) with a homogeneous spectrumE (see Definition 3.2) and with diagonal elements of the resolventR(m, m; z) having pure imaginary boundary values a.e. onE. 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 classJ E 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 spaceH 2. 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 classJ E .