On Mbekhta's conjecture and pseudo-Fredholm operators on Banach and Hilbert spaces

@misc{aznay2026mbekhtas,
  title = {{On Mbekhta's conjecture and pseudo-Fredholm operators on Banach and Hilbert spaces}},
  author = {Zakariae Aznay and Fadil Chabbabi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23229},
  eprint = {2610.23229},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23229},
}

TY  - UNPB
TI  - On Mbekhta's conjecture and pseudo-Fredholm operators on Banach and Hilbert spaces
AU  - Zakariae Aznay
AU  - Fadil Chabbabi
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.23229
UR  - https://arxiv.org/abs/2610.23229
AB  - We give negative answers to Mbekhta's conjecture \cite[p.~261]{Mbekhta} and to a related conjecture on pseudo-B-Fredholm operators \cite[p.~202]{AZO}. A sufficient condition for pseudo-Fredholmness on Banach spaces is established. It yields a characterization on Hilbert spaces in terms of the closedness of cc(T), cc(T)+qn(T) and Ran(T)+qn(T), as well as characterizations of pseudo-B-Fredholm and pseudo-B-Weyl operators on Banach spaces. A counterexample on a separable Banach space shows that the three closedness conditions do not characterize pseudo-Fredholmness in general. We also construct two operators having the same analytic core, quasinilpotent part, and kernels and ranges of all powers, although only one is pseudo-Fredholm. Moreover, a semi-B-Fredholm operator on a Banach space need not be a direct sum of a semi-Fredholm operator and a nilpotent operator, answering the question raised in \cite[p.~113]{AznayKato2023}. Finally, the three closedness conditions in the Hilbert-space characterization are shown to be independent.
ER  -