Generalized Convexity, Blaschke-type Condition in Unbounded Domains, and Application in Spectral Perturbation Theory of Linear Operators

Seminar
Speaker
Prof. S. Favorov, Kharkov University, Ukraine
Date
08/04/2013 - 14:00Add to Calendar 2013-04-08 14:00:00 2013-04-08 14:00:00 Generalized Convexity, Blaschke-type Condition in Unbounded Domains, and Application in Spectral Perturbation Theory of Linear Operators We introduce a notion of r-convexity for subsets of the complex plane. It is a pure geometric characteristic that generalizes the         usual notion of convexity. Next, we investigate subharmonic               functions that grow near the boundary in unbounded domains with           r-convex compact complement. We obtain the Blaschke-type bounds           for its Riesz measure and, in particular, for zeros of unbounded          analytic functions  in unbounded domains. These results are based         on a certain estimates for Green functions on complements of some         neighborhoods of $r$-convex compact set. Also, we apply our               results in perturbation theory of linear operators in a Hilbert           space. More precisely, we find quantitative estimates for the rate        of condensation of the discrete spectrum of a perturbed operator          near its the essential spectrum.                                          אוניברסיטת בר-אילן - Department of Mathematics mathoffice@math.biu.ac.il Asia/Jerusalem public
Abstract

We introduce a notion of r-convexity for subsets of the complex
plane. It is a pure geometric characteristic that generalizes the        
usual notion of convexity. Next, we investigate subharmonic              
functions that grow near the boundary in unbounded domains with          
r-convex compact complement. We obtain the Blaschke-type bounds          
for its Riesz measure and, in particular, for zeros of unbounded         
analytic functions  in unbounded domains. These results are based        
on a certain estimates for Green functions on complements of some        
neighborhoods of $r$-convex compact set. Also, we apply our              
results in perturbation theory of linear operators in a Hilbert          
space. More precisely, we find quantitative estimates for the rate       
of condensation of the discrete spectrum of a perturbed operator         
near its the essential spectrum.                                         

Last Updated Date : 03/04/2013