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

TL;DR

This paper establishes an optimal Blaschke-type condition for the Riesz measure of subharmonic functions with specific growth near a closed set on the unit circle, with applications to contraction operators close to unitaries.

Contribution

It introduces a new Blaschke-type condition for subharmonic functions with growth constraints, extending previous results and applying to contraction operators near unitaries.

Findings

Derived an optimal Blaschke-type condition for Riesz measures.

Extended results to cases where the growth set is finite.

Applied the theoretical findings to contraction operators close to unitaries.

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.