We obtain a Blaschke-type necessary condition on zeros of analytic functions on the unit disc with different types of exponential growth at the boundary. These conditions are used to prove Lieb-Thirring-type inequalities for the eigenvalues of complex Jacobi matrices.
Let p be a trigonometric polynomial, non-negative on the unit circle T. We say that a measure on T belongs to the polynomial Szegő class, if d (e i ) = ac (e i ) d + d s (e i ), s is singular, and 2 0 p(e i ) log ac (e i ) d > − ∞.For the associated orthogonal polynomials { n }, we obtain pointwise asymptotics inside the unit disc D. Then we show that these asymptotics hold in L 2 -sense on the unit circle. As a corollary, we get an existence of certain modified wave operators.
We obtain various versions of classical Lieb-Thirring bounds for one-and multi-dimensional complex Jacobi matrices. Our method is based on Fan-Mirski Lemma and seems to be fairly general.
This is a sequel of the article by Borichev-Golinskii-Kupin [2] where the authors obtain Blaschke-type conditions for special classes of analytic functions in the unit disk which satisfy certain growth hypotheses. These results were applied to get Lieb-Thirring inequalities for complex compact perturbations of a selfadjoint operator with a simply connected resolvent set.The first result of the present paper is an appropriate local version of the Blaschke-type condition from [2]. We apply it to obtain a similar condition for an analytic function in a finitely connected domain of a special type. Such condition is by and large the same as a Lieb-Thirring type inequality for complex compact perturbations of a selfadjoint operator with a finite-band spectrum. A particular case of this result is the Lieb-Thirring inequality for a selfadjoint perturbation of the Schatten class of a periodic (or a finite-band) Jacobi matrix. The latter result seems to be new in such generality even in this framework.
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