Eigenvalue estimates of positive integral operators with analytic kernels
type: Item
instance of: scholarly article
title: Eigenvalue estimates of positive integral operators with analytic kernels
zbMATH Open document ID: 1205.45003
author: Y. Soykan
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 2010-11-03T00:00:00Z
review text: Suppose that \(\Omega\) is a simply connected domain with an infinite complement, \(\Delta\) is the unit disc, \(\varphi : \Omega \rightarrow \Delta \) is a conformal map, \[ K_{\Omega} (z, \zeta)= \frac{\left(\varphi'(\zeta)\right)^{\frac{1}{2}} \overline{\left(\varphi'(z)\right)^{\frac{1}{2}}}}{1-\varphi(\zeta)\overline{\varphi(z)}}. \] Suppose that \(J\) is a closed real interval such that \(J \subset \Omega\) and \[ (T_{\Omega}f)(s) = \int_{J} K_{\Omega}(s, t) f(t)dt, \;s\in J. \] The operator \(T_{\Omega}\) is compact and positive (\((T_{\Omega} f, f) \geq 0\)) in the space \(L_2(J)\). Such operators studied by \textit{G. Little} in his papers [Math. Proc. Camb. Philos. Soc. 91, 267--284 (1982; Zbl 0489.45001), Proc. Lond. Math. Soc., III. Ser. 62, No.~2, 403--426 (1991; Zbl 0682.45004)]. Let \(C=(C_1, \dots , C_N) \) be an \(N\)-tuple of conic sections. Let \(D\) be a complementary domain of \(\bigcup_{i=1}^N C_i\), and suppose that \(D\) is simply connected. Let \(D_i\) be the complementary domain of \(C_i\) which contains \(D\). The main result of the author states that there exists some constant \(m>0\) such that \( m T_D \leq T_{D_1} +\cdots + T_{D_N}\).
Mathematics Subject Classification ID: 45C05
Mathematics Subject Classification ID: 30C40
Mathematics Subject Classification ID: 45P05
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47G10
zbMATH DE Number: 5810051
zbMATH Keywords: eigenvalue problems
zbMATH Keywords: special kernels
zbMATH Keywords: positive integral operators
zbMATH Keywords: Hardy spaces
zbMATH Keywords: Smirnov classes
zbMATH Keywords: closed analytic curves
zbMATH Keywords: Ahlfors regular curves
zbMATH Keywords: eigenvalue estimates
zbMATH Keywords: analytic kernels
zbMATH Keywords: conformal map
MaRDI profile type: Publication
full work available at URL: s12044-010-0032-1
OpenAlex ID: W1963581542
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DOI: 10.1007/S12044-010-0032-1
description: scientific article; zbMATH DE number 5810051