TY - JOUR

T1 - Almost periodic Jacobi matrices with homogeneous spectrum, infinite dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions

AU - Sodin, Mikhail

AU - Yuditskii, Peter

PY - 1997

Y1 - 1997

N2 - 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.

AB - 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.

KW - Almost periodic Jacobi matrices

KW - Character-automorphic functions

KW - Generalized Abel map

KW - Jacobi inversion

KW - Widom type Fuchsian groups

UR - http://www.scopus.com/inward/record.url?scp=29344445693&partnerID=8YFLogxK

U2 - 10.1007/bf02921627

DO - 10.1007/bf02921627

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AN - SCOPUS:29344445693

SN - 1050-6926

VL - 7

SP - 387

EP - 435

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

IS - 3

ER -