A Blaschke-type condition for analytic and subharmonic functions and application to contraction operators

Abstract

Let $E$ be a closed set on the unit circle. We find a Blaschke-type condition, optimal in a sense of the order, on the Riesz measure of a subharmonic function $v$ in the unit disk with a certain growth at the direction of $E$. In particular case when $E$ is a finite set, and $v=\log |f|$ with an analytic function $f$, our result agrees with the recent one by A. Borichev, L. Golinskii and S. Kupin. An application to contractions close to unitary operators in the Hilbert space is given.

12 pages, Bibliography 19 items. Added references, corrected typos