## 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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