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

This is a sequel of the article by Borichev, Golinskii and Kupin (2009) [1], 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 et al. (2009) [1]. 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 finite-band) Jacobi matrix.