Meromorphic generalized inverses of operator functions
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.
inversion of holomorphic operator functions
scientific article; zbMATH DE number 6112939
scientific article