Drazin inverses of operators with rational resolvent. (English) Zbl 1199.47021

Schmoeger, Christoph

Publ. Inst. Math., Nouv. Sér. 83(97), 37-47 (2008).

An operator \(A\) is said to have a rational resolvent if for \(\lambda\) in the resolvent set \(\rho(A)=\mathbb C\smallsetminus\sigma(A)\) of \(A\), the resolvent \((A-\lambda I)^{-1}\) can be expressed as \(P(\lambda)/q(\lambda)\), where \(P(\lambda)\) is a polynomial with coefficients in the algebra of all bounded linear operators on \(X\) and \(q(\lambda)\) is a complex polynomial such that \(P\) and \(q\) have no common zero. The author studies spectral properties of the Drazin inverse of operators with rational resolvents. He proves that if \(\lambda_0\in\sigma(A)\) and \(C\) is the Drazin inverse of \(A-\lambda_0\), then there is a scalar polynomial \(p\) such that \(C=p(A)\).

Reviewer: Mohammad Sal Moslehian (Mashhad)

MSC:

47A10	Spectrum, resolvent
47A05	General (adjoints, conjugates, products, inverses, domains, ranges, etc.)