We study self-adjoint bounded Jacobi operators of the form:(Jψ)(n) = a n ψ(n + 1) + b n ψ(n) + a n−1 ψ(n − 1) on ℓ 2 (N). We assume that for some fixed q ∈ N, the q-variation of {a n } and {b n } is square-summable and {a n } and {b n } converge to q-periodic sequences {a per n } and {b per n }, respectively. Our main result is that under these assumptions the essential support of the absolutely continuous part of the spectrum of J is equal to that of the asymptotic periodic Jacobi operator.This work generalizes a recent result of S. A. Denisov.

We consider random Jacobi matrices of the form (J ω u)(n) = a n (ω)u(n + 1) + b n (ω)u(n) + a n−1 (ω)u(n − 1) on ℓ 2 (N), where a n (ω) =ã n + α n (ω), b n (ω) =b n + β n (ω), {ã n } and {b n } are sequences of bounded variation obeyingã n → 1 and b n → 0, and {α n (ω)} and {β n (ω)} are sequences of independent random variables on a probability space (Ω, dP (ω)) obeying ∞ n=1 Ω (α 2 n (ω) + β 2 n (ω)) dP (ω) < ∞ and Ω α n (ω) dP (ω) = Ω β n (ω) dP (ω) = 0 for each n. We further assume that there exists C 0 > 0 such that 1/C 0 < a n (ω) < C 0 and −C 0 < b n (ω) < C 0 for every n and P a.e. ω. We prove that, for P a.e. ω, J ω has purely absolutely continuous spectrum on (−2, 2).

We consider random self-adjoint Jacobi matrices of the form (J ω u)(n) = a n (ω)u(n + 1) + b n (ω)u(n) + a n−1 (ω)u(n − 1) on 2 (N), where {a n (ω) > 0} and {b n (ω) ∈ R} are sequences of random variables on a probability space (Ω, dP (ω)) such that there exists q ∈ N, such that for any l ∈ N,are independent random variables of zero mean satisfying ∞ n=1 Ω β 2 n (ω) dP (ω) < ∞.Let J p be the deterministic periodic (of period q) Jacobi matrix whose coefficients are the mean values of the corresponding entries in J ω . We prove that for a.e. ω, the a.c. spectrum of the operator J ω equals to and fills the spectrum of J p . If, moreover,then for a.e. ω, the spectrum of J ω is purely absolutely continuous on the interior of the bands that make up the spectrum of J p .

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