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Meromorphic generalized inverses of operator functions. (English) Zbl 1272.47019
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 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 left (resp., right) 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 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.

MSC:
47A53 (Semi-) Fredholm operators; index theories
47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
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