Algebraic Direct Sums and Generalized Drazin Invertibility

Abstract

Let T be a bounded linear operator on a complex Banach space X. We prove that the algebraic decomposition X=K(T)⊕ Hq(T) forces both summands to be closed and hence characterizes generalized Drazin invertibility. This improves the classical characterizations of Mbekhta~\resultcite{Theorem~1.6}{MbekhtaKato}, Schmoeger~\resultcite{Theorem~4}{Schmoeger}, and Dajić and Koliha~\resultcite{Theorem~6.7}{DajicKoliha}, by removing their closedness assumptions. We also show that T is generalized Drazin invertible if and only if dim(K(T)∩ Hq(T))<∞ and codim(K(T)+Hq(T))<∞. For ordinary Drazin invertibility, we obtain analogous finite-defect characterizations involving the hyperkernel and the hyperrange. We also give a characterization using pointwise nilpotence on an invariant summand. Finally, we consider arbitrary invariant algebraic decompositions and give a counterexample when the inverse of the restriction to one summand is not assumed bounded.

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BibTeX RIS

BibTeX

@misc{aznay2026algebraic,
  title = {{Algebraic Direct Sums and Generalized Drazin Invertibility}},
  author = {Zakariae Aznay and Fadil Chabbabi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18433},
  eprint = {2610.18433},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18433},
}

RIS

TY  - UNPB
TI  - Algebraic Direct Sums and Generalized Drazin Invertibility
AU  - Zakariae Aznay
AU  - Fadil Chabbabi
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.18433
UR  - https://arxiv.org/abs/2610.18433
AB  - Let T be a bounded linear operator on a complex Banach space X. We prove that the algebraic decomposition X=K(T)⊕ Hq(T) forces both summands to be closed and hence characterizes generalized Drazin invertibility. This improves the classical characterizations of Mbekhta~\resultcite{Theorem~1.6}{MbekhtaKato}, Schmoeger~\resultcite{Theorem~4}{Schmoeger}, and Dajić and Koliha~\resultcite{Theorem~6.7}{DajicKoliha}, by removing their closedness assumptions. We also show that T is generalized Drazin invertible if and only if dim(K(T)∩ Hq(T))<∞ and codim(K(T)+Hq(T))<∞. For ordinary Drazin invertibility, we obtain analogous finite-defect characterizations involving the hyperkernel and the hyperrange. We also give a characterization using pointwise nilpotence on an invariant summand. Finally, we consider arbitrary invariant algebraic decompositions and give a counterexample when the inverse of the restriction to one summand is not assumed bounded.
ER  -

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