Meromorphic generalized inverses of operator functions
type: Item
instance of: scholarly article
title: Meromorphic generalized inverses of operator functions
zbMATH Open document ID: 1272.47019
published in: Indagationes Mathematicae. New Series
publication date: 2012-12-06T00:00:00Z
review text: Let \(X,Y\) be Banach spaces and let \(L(X,Y)\) denote the Banach space of all bounded linear operators acting from \(X\) into \(Y\). An operator \(T\in L(X,Y)\) is called \textit{relatively regular} (\(T\in\mathcal{R}(X,Y)\)) if it has a generalized inverse \(U\in L(Y,X)\) defined by the properties \(TUT=T\) and \(UTU=U\). An operator \(T\in\mathcal{R}(X,Y)\) is called a \textit{left} (resp., \textit{right}) \textit{semi-Fredholm} operator if its kernel (resp., its cokernel) is finite-dimensional. If a holomorphic operator function \(T\) has the inverse meromorphic function \(T^{-1}\) admitting local decompositions \[ T(\lambda)^{-1}=A(\lambda)+\frac{1}{f(\lambda)}\,B(\lambda)\,F\, C(\lambda), \] where \(A,B,C\) are holomorphic operator functions, \(f\not\equiv 0\) is a scalar holomorphic function and \(F\) is an operator of finite rank, then the meromorphic function \(T^{-1}\) is called \textit{finite-meromorphic}. The paper is a survey of results concerning the invertibility of holomorphic operator functions. The existence of finite-meromorphic generalized inverses for holomorphic semi-Fredholm operator functions is studied. Further, the existence of smooth finite-meromorphic generalized inverses is investigated for semi-Fredholm or relatively regular holomorphic functions of one or several complex variables. Global decompositions of finite-meromorphic operator functions are constructed and applications to vector and operator function equations are considered.
reviewed by: Yuri I. Karlovich
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 47A56
Mathematics Subject Classification ID: 47A05
zbMATH DE Number: 6112939
zbMATH Keywords: Fredholm operator
zbMATH Keywords: generalized inverse
zbMATH Keywords: meromorphic operator function
zbMATH Keywords: inversion of holomorphic operator functions
author: Winfried Kaballo
MaRDI profile type: Publication
full work available at URL: j.indag.2012.05.001
OpenAlex ID: W1969202231
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Meromorphic generalized inverses of operator functions (DOI)
DOI: 10.1016/J.INDAG.2012.05.001

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