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

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 \(C=p(A)\).

MSC:

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