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The Drazin inverses of sum and difference of idempotents. (English) Zbl 1161.47001

The paper is concerned with the Drazin inverses \((P \pm Q)^{D}\) of the sum and difference of two idempotents \(P\) and \(Q\), a problem that was first considered by M. P. Drazin in [Am. Math. Mon. J. 65, 506–514 (1958; Zbl 0083.02901)]. Using the technique of operator block matrices, the author presents some formulas for \((P \pm Q)^{D}\) in different cases: (i) \(PQP=O\); (ii) \(PQP=PQ\); (iii) \(PQP=P\).

MSC:

47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
46C07 Hilbert subspaces (= operator ranges); complementation (Aronszajn, de Branges, etc.)
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
15A09 Theory of matrix inversion and generalized inverses

Citations:

Zbl 0083.02901
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Full Text: DOI

References:

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