Revista Matemática Iberoamericana


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Volume 31, Issue 1, 2015, pp. 1–32
DOI: 10.4171/RMI/824

Published online: 2015-03-05

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

Sergey Favorov[1] and Leonid B. Golinskii[2]

(1) Karazin Kharkov National University, Ukraine
(2) Institute for Low Temperature Physics, Kharkov, Ukraine

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.

Keywords: Subharmonic functions, Riesz measure, Riesz decomposition theorem, Green’s function, Schatten–von Neumann operators

Favorov Sergey, Golinskii Leonid: Blaschke-type conditions on unbounded domains, generalized convexity, and applications in perturbation theory. Rev. Mat. Iberoamericana 31 (2015), 1-32. doi: 10.4171/RMI/824