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

All three subjects reflected in the title are closely intertwined in the paper. Let J_E 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 J_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 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 J_E.