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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.

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.

Reviewer: D.Przeworska-Rolewicz (Warszawa)

### MSC:

47A05 | General (adjoints, conjugates, products, inverses, domains, ranges, etc.) |

15A09 | Theory of matrix inversion and generalized inverses |

### Keywords:

regular element; spectral element; Drazin inverse; quasinilpotent elements; Banach algebras with unit; isolated spectral point; differential equations in Banach spaces; nilpotent elements
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\textit{J. J. Koliha}, Glasg. Math. J. 38, No. 3, 367--381 (1996; Zbl 0897.47002)

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