# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  Taylor, Introduction to functional analysis (1980) · Zbl 0501.46003  Harte, Invertibility and Singularity for Bounded Linear Operators (1988)  DOI: 10.1137/0131057 · Zbl 0355.15008  Harte, Irish Math. Soc. Newsletter 11 pp 10– (1984)  DOI: 10.1016/0024-3795(92)90237-5 · Zbl 0743.15006  DOI: 10.2307/2308576 · Zbl 0083.02901  Caradus, Generalized inverses and operator theory (1978) · Zbl 0434.47003  DOI: 10.1137/0131035 · Zbl 0341.34001  Campbell, Generalized Inverses of Linear Transformations (1979)  Campbell, Recent Applications of Generalized Inverses pp 250– (1982)  Bouldin, Recent Applications of Generalized Inverses pp 233– (1982)  Ben-Israel, Generalized Inverses: Theory and Applications (1974)  Nashed, World Sci. Ser. Appl. Anal. 1 pp 441– (1992)  Nashed, Generalized Inverses and Applications (1976)  Marek, Matrix Analysis for Applied Sciences, Vol. 2 84 (1986) · Zbl 0613.15002  DOI: 10.1137/0129011 · Zbl 0309.47014  King, Pacific J. Math. 70 pp 383– (1977) · Zbl 0382.47001  Heuser, Functional Analysis pp New York– (1982)  Hartwig, Pacific J. Math. 78 pp 133– (1978) · Zbl 0362.15003  Harte, PanAm. Math. J. 1 pp 10– (1991)  DOI: 10.1007/BF01388703 · Zbl 0797.65054
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.