Remarks on the range and the kernel of generalized derivation

  • Y. Bouhafsi Chouaib Doukkali University, Faculty of Science, Department of Mathematics El Jadida, Morocco D´epartement de Math´ematiques Centre R´egional des M´etiers de l’Education et de la Formation Marrakech-Safi, Marocco
  • M. Ech-chad Ibn Tofail University, Faculty of Science K´enitra, Morocco
  • M. Missouri Ibn Tofail University, Faculty of Science K´enitra, Morocco
  • A. Zouaki Ibn Tofail University, Faculty of Science K´enitra, Morocco
Keywords: derivation; orthogonality; Fuglede-Putnam property; subnormal operator; compact operator; dominant operator

Abstract

Let $L(H)$ denote the algebra of operators on a complex
infinite dimensional Hilbert space $H$ and let $\;\mathcal{J}$
denote a two-sided ideal in $L(H)$. Given $A,B\in L(H)$, define
the generalized derivation $\delta_{A,B}$ as an operator on
$L(H)$ by

\centerline{$\delta_{A,B}(X)=AX-XB.$}

\smallskip\noi We say that the pair of
operators $(A,B)$ has the Fuglede-Putnam property
$(PF)_{\mathcal{J}}$ if $AT=TB$ and $T\in \mathcal{J}$ implies
$A^{\ast}T=TB^{\ast}$. In this paper, we give operators $A,B$ for
which the pair $(A,B)$ has the property $(PF)_{\mathcal{J}}$. We
establish the orthogonality of the range and the kernel of a
generalized derivation $\delta_{A,B}$ for non-normal operators $A,
B\in L(H)$. We also obtain new results concerning the intersection
of the closure of the range and the kernel of $\delta_{A,B}$.

Author Biographies

Y. Bouhafsi, Chouaib Doukkali University, Faculty of Science, Department of Mathematics El Jadida, Morocco D´epartement de Math´ematiques Centre R´egional des M´etiers de l’Education et de la Formation Marrakech-Safi, Marocco

Chouaib Doukkali University, Faculty of Science, Department of Mathematics
El Jadida, Morocco
D´epartement de Math´ematiques
Centre R´egional des M´etiers de l’Education et de la Formation
Marrakech-Safi, Marocco

M. Ech-chad, Ibn Tofail University, Faculty of Science K´enitra, Morocco

Ibn Tofail University, Faculty of Science
K´enitra, Morocco

M. Missouri, Ibn Tofail University, Faculty of Science K´enitra, Morocco

Ibn Tofail University, Faculty of Science
K´enitra, Morocco

A. Zouaki, Ibn Tofail University, Faculty of Science K´enitra, Morocco

Ibn Tofail University, Faculty of Science
K´enitra, Morocco

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Published
2022-06-27
How to Cite
Bouhafsi, Y., Ech-chad, M., Missouri, M., & Zouaki, A. (2022). Remarks on the range and the kernel of generalized derivation. Matematychni Studii, 57(2), 202-209. https://doi.org/10.30970/ms.57.2.202-209
Section
Articles