Blaschke-type conditions on unbounded domains, generalized convexity, and applications in perturbation theory

Abstract

We introduce a new geometric characteristic of compact sets in the plane called $r$-convexity, which fits nicely into the concept of generalized convexity and extends standard convexity in an essential way. We obtain a Blaschke-type condition for the Riesz measures of certain subharmonic functions on unbounded domains with $r$-convex complements, having growth governed by the distance to the boundary. The result is applied to the study of the convergence of the discrete spectrum for the Schatten–von Neumann perturbations of bounded linear operators in Hilbert space.