Abstract
Let $\mathcal{L}(H)$ denote the algebra of all bounded
linear operators on a complex infinite dimensional Hilbert space
$H$ into itself. Given $A,B\in \mathcal{L}(H)$, the generalized
derivation $\delta_{A,B}\in \mathcal{L}(\mathcal{L}(H))$ is
defined by $\delta_{A,B}(X)=AX-XB$. For hyponormal operators $A$
and $B$, L.A. Fialkow gave necessary conditions for which the
range of $\delta_{A,B}$ is norm closed in $\mathcal{L}(H)$ [13].
In this paper, we present necessary or sufficient conditions under
which the generalized derivation $\delta_{A,B}$ has closed range,
where $A$ and $B$ are $p$-hyponormal operators. We give a
characterization for the range of $\delta_{A,B}$ to have closed
range in the case when $A$ and $B$ are cyclic subnormal operators.
Key words: Derivation, Generalized derivation, Generalized inverse,
Subnormal operator, $p$-hyponormal operator, $w$-hyponormal
operator.
Full Text
Article Information
| Title | On the Range of a Generalized Derivation |
| Source | Methods Funct. Anal. Topology, Vol. 32 (2026), no. 2, 129-138 |
| DOI | 10.31392/MFAT-npu26_2.2026.03 |
| Milestones | Received 01/08/2025; Revised 31/12/2025 |
| Copyright | The Author(s) 2026 (CC BY-SA) |
Authors Information
Hamza El Mouadine
Laboratoire LIMA D´epartement de Mathematiques, Faculte des Sciences, Universit´e Chouaib Doukkali, Avenue Jabran Khalil Jabran, P. O. Box 20, El Jadida, Maroc
Abdelkhalek Faouzi
Laboratoire LIMA D´epartement de Mathematiques, Faculte des Sciences, Universit´e Chouaib Doukkali, Avenue Jabran Khalil Jabran, P. O. Box 20, El Jadida, Maroc
Youssef Bouhafsi
Laboratoire LIMA D´epartement de Mathematiques, Faculte des Sciences, Universit´e Chouaib Doukkali, Avenue Jabran Khalil Jabran, P. O. Box 20, El Jadida, Maroc.
Permanent Address: Departement de Mathematiques, Centre Regional des Metiers de l’Education et de la Formation Marrakech-Safi, Safi, Maroc.
Citation Example
Hamza El Mouadine, Abdelkhalek Faouzi, and Youssef Bouhafsi, On the Range of a Generalized Derivation, Methods Funct. Anal. Topology 32
(2026), no. 2, 129-138.
BibTex
@article {MFAT2278,
AUTHOR = {Hamza El Mouadine and Abdelkhalek Faouzi and Youssef Bouhafsi},
TITLE = {On the Range of a Generalized Derivation},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {32},
YEAR = {2026},
NUMBER = {2},
PAGES = {129-138},
ISSN = {1029-3531},
DOI = {10.31392/MFAT-npu26_2.2026.03},
URL = {https://mfat.imath.kiev.ua/article/?id=2278},
}
