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A generalized Drazin inverse. (English) Zbl 0897.47002
Let $$a$$, $$b$$ be elements of a semigroup. Then $$b$$ is a Drazin inverse of $$a$$ (written $$b= a^d$$) if $$ab= ba$$, $$b= ab^2$$, $$a^k= a^{k+ 1}b$$ for some nonnegative integer $$k$$. An element $$a$$ of a ring $$A$$ is quasinilpotent if, for every $$x$$ commuting with $$a$$, the element $$e-xa$$ is invertible, where $$e$$ is the unit of $$A$$. The set of all quasinilpotent elements in $$A$$ is denoted by $$QN(A)$$. Let $$a\in A$$. Then $$b\in A$$ is a Drazin inverse $$b= a^D$$ of $$a$$ if $$ab= ba$$, $$b= ab^2$$, $$a- a^2b\in QN(A)$$. This definition is used also in the case of Banach algebras with unit, since in that case $$QN(A)$$ is the set of all elements $$x$$ such that $$\| x^n\|^{1/n}\to 0$$ as $$n\to\infty$$.
In the paper under review, there are studied mainly Drazin inverses for bounded linear operators in a Banach space $$X$$ when $$0$$ is an isolated spectral point of an operator. These results are applied in order to solve differential equations in Banach spaces. However, this paper can also be considered as a survey of the subject. Slightly misleading is that through the paper $$N(A)$$ denotes the set of all nilpotent elements in a ring $$A$$, although in Section 7 by the same $$N(A)$$ is meant the null space of an operator $$A\in L(X)$$, where $$X$$ is a Banach space.

##### MSC:
 47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.) 15A09 Theory of matrix inversion and generalized inverses
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