A generalized Drazin inverse. (English) Zbl 0897.47002

Koliha, J. J.

Glasg. Math. J. 38, No. 3, 367-381 (1996).

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.

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

References:

[1]	Taylor, Introduction to functional analysis (1980) · Zbl 0501.46003
[2]	Harte, Invertibility and Singularity for Bounded Linear Operators (1988)
[3]	DOI: 10.1137/0131057 · Zbl 0355.15008
[4]	Harte, Irish Math. Soc. Newsletter 11 pp 10– (1984)
[5]	DOI: 10.1016/0024-3795(92)90237-5 · Zbl 0743.15006
[6]	DOI: 10.2307/2308576 · Zbl 0083.02901
[7]	Caradus, Generalized inverses and operator theory (1978) · Zbl 0434.47003
[8]	DOI: 10.1137/0131035 · Zbl 0341.34001
[9]	Campbell, Generalized Inverses of Linear Transformations (1979)
[10]	Campbell, Recent Applications of Generalized Inverses pp 250– (1982)
[11]	Bouldin, Recent Applications of Generalized Inverses pp 233– (1982)
[12]	Ben-Israel, Generalized Inverses: Theory and Applications (1974)
[13]	Nashed, World Sci. Ser. Appl. Anal. 1 pp 441– (1992)
[14]	Nashed, Generalized Inverses and Applications (1976)
[15]	Marek, Matrix Analysis for Applied Sciences, Vol. 2 84 (1986) · Zbl 0613.15002
[16]	DOI: 10.1137/0129011 · Zbl 0309.47014
[17]	King, Pacific J. Math. 70 pp 383– (1977) · Zbl 0382.47001
[18]	Heuser, Functional Analysis pp New York– (1982)
[19]	Hartwig, Pacific J. Math. 78 pp 133– (1978) · Zbl 0362.15003
[20]	Harte, PanAm. Math. J. 1 pp 10– (1991)
[21]	DOI: 10.1007/BF01388703 · Zbl 0797.65054