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On one condition of absolutely continuous spectrum for self-adjoint operators and its applications

Ianovich E.

Opuscula Mathematica

2018

Vol. 38, no. 5

699--718

In this work the method of analyzing of the absolutely continuous spectrum for self-adjoint operators is considered. For the analysis it is used an approximation of a self-adjoint operator A by a sequence of operators An with absolutely continuous spectrum on a given interval [a, b] which converges to A in a strong sense on a dense set. The notion of equi-absolute continuity is also used. It was found a sufficient condition of absolute continuity of the operator A spectrum on the finite interval [a, b] and the condition for that the corresponding spectral density belongs to the class Lp[a,b] (p ≥ 1). The application of this method to Jacobi matrices is considered. As one of the results we obtain the following assertion: Under some mild assumptions, suppose that there exist a constant C > 0 and a positive function g(x) ∈ Lp[a, b] (p ≥ 1).such that for all n sufficiently large and almost all [formula] the estimate [formula] holds, where Pn(x) are 1st type polynomials associated with Jacobi matrix (in the sense of Akhiezer) and bn is a second diagonal sequence of Jacobi matrix. Then the spectrum of Jacobi matrix operator is purely absolutely continuous on [a, b] and for the corresponding spectral density ƒ (x) we have ƒ (x) ∈ Lp[a,b].