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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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