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Type of Document	Dissertation
Author	Zlatos, Andrej
Author's Email Address	andrej AT caltech.edu
URN	etd-05222003-114151
Persistent URL	http://resolver.caltech.edu/CaltechETD:etd-05222003-114151
Title	Sum rules and the Szego condition for Jacobi matrices
Degree	PhD
Option	Mathematics
Advisory Committee	Advisor Name Title Barry Simon Committee Chair Nikolai Makarov Committee Member Rowan Killip Committee Member Wilhelm Schlag Committee Member
Keywords	spectral theory sum rules jacobi matrices szego condition
Date of Defense	2003-05-12
Availability	unrestricted
Abstract We consider Jacobi matrices J with real b_n on the diagonal, positive a_n on the next two diagonals, and with u'(x) the density of the absolutely continuous part of the spectral measure. In particular, we are interested in compact perturbations of the free matrix J_0, that is, such that the a_n go to 1 and b_n go to 0. We study the Case sum rules for such matrices. These are trace formulae relating sums involving the a_n's and b_n's on one side and certain quantities in terms of the spectral measure on the other. We establish situations where the sum rules are valid, extending results of Case and Killip-Simon. The matrix J is said to satisfy the Szego condition whenever the integral int_{0}^{pi} log [u'(2 cos t)] dt, which appears in the sum rules, is finite. Applications of our results include an extension of Shohat's classification of certain Jacobi matrices satisfying the Szego condition to cases with an infinite point spectrum, and a proof that if n(a_n - 1) go to a, nb_n go to b, and 2a < |b|, then the Szego condition fails. Related to this, we resolve a conjecture by Askey on the Szego condition for Jacobi matrices which are Coulomb perturbations of J_0. More generally, we prove that if a_n = 1 + a/n^c + O(n^{-1-eps}) and b_n = b/n^c + O(n^{-1-eps}) with 0 < c <= 1 and eps > 0, then the Szego condition is satisfied if and only if 2a >= |b|
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