SUM RULES FOR JACOBI MATRICES AND THEIR APPLICATIONS TO SPECTRAL THEORY

Abstract

The study discusses the proof of and symmetric application of Cases sum rules for Jacobi matrices. Of special interest is a linear combination of these sum rules which have strictly positive terms. The complete classification of the spectral measure of all Jacobi matrices J for which  J-J0 is Hilbert space -Achmidt. The study shows the bound of a Jacobi matrix. The description for the point and absolutely continuous spectrum, while for the singular continuous spectrum additional assumptions are needed.  The study shows and prove a bound of a Jacobi matrix. And we give complete description for the point and absolutely continuous spectrum, while for the singular continuous spectrum additional assumptions are needed, we prove a characterization of a characteristic function of a row contraction operator and verify its defect operator. We also prove a commutability of an operator of this row contraction.