A Blaschke-type condition for analytic functions on finitely connected domains. Applications to complex perturbations of a finite-band selfadjoint operator.

Leonid Golinskii (1) , Stanislas Kupin (2)

1 Mathematics, ILTPE
2 IMB - Institut de Mathématiques de Bordeaux

This is a sequel of the article by Borichev-Golinskii-Kupin [2009], 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 Borichev-Golinskii-Kupin [2009]. 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.

Mots clés: Lieb-Thirring inequality; Blaschke-type condition; finite-band selfadjoint operators and their complex perturbations.; finite-band selfadjoint operators and their complex perturbations

Domaines: Physique mathématique [math-ph]; Théorie spectrale [math.SP]

Leonid Golinskii, Stanislas Kupin. A Blaschke-type condition for analytic functions on finitely connected domains. Applications to complex perturbations of a finite-band selfadjoint operator.. Journal of Mathematical Analysis and Applications, 2012, 389 (2), pp.705-712. ⟨hal-00781336⟩