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Perturbations of discrete spectra of holomorphic operator-valued functions

  • Rafikul AlamEmail author
  • Jibrail Ali
S.I.: ICWAA-2018
  • 21 Downloads

Abstract

Let X be a complex Banach space and L(X) be the Banach space of all bounded linear operators on X. Let \(\Omega \subset {{\mathbb {C}}}\) be open and connected. Let \(T, V : \Omega \longrightarrow L(X)\) be holomorphic operator-valued functions. We consider the one parameter family of operator-valued functions \(W(\lambda , t) := T(\lambda ) + t V(\lambda )\), for \(t \in {{\mathbb {C}}}\), and analyze evolution of the discrete eigenvalues of \(W(\lambda , t)\) when t varies in \({{\mathbb {C}}}.\) We provide a brief review of the discrete spectrum of \(T(\lambda )\) and present several equivalent characterizations for discrete eigenvalues of \(T(\lambda ).\) We also prove Rouche’s theorem for operator-valued functions under a weaker assumption, which we utilize to derive perturbation bounds for the discrete eigenvalues of \(W(\lambda , t)\) when |t| is small.

Keywords

Banach space Fredholm operator Operator-valued function Spectrum Discrete spectrum Discrete eigenvalues Linearization 

Mathematics Subject Classifications

47A75 47A55 47A10 47A53 

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

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Copyright information

© Forum D'Analystes, Chennai 2019

Authors and Affiliations

  1. 1.Department of MathematicsIIT GuwahatiGuwahatiIndia

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