On the range of some elementary operators

Hamza El Mouadine; Abdelkhalek Faouzi; Youssef Bouhafsi

Commentationes Mathematicae Universitatis Carolinae (2024)

  • Issue: 1, page 53-62
  • ISSN: 0010-2628

Abstract

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Let L ( H ) denote the algebra of all bounded linear operators on a complex infinite dimensional Hilbert space H . For A , B ∈ L ( H ) , the generalized derivation δ A , B and the multiplication operator M A , B are defined on L ( H ) by δ A , B ( X ) = A X - X B and M A , B ( X ) = A X B . In this paper, we give a characterization of bounded operators A and B such that the range of M A , B is closed. We present some sufficient conditions for δ A , B to have closed range. Some related results are also given.

How to cite

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El Mouadine, Hamza, Faouzi, Abdelkhalek, and Bouhafsi, Youssef. "On the range of some elementary operators." Commentationes Mathematicae Universitatis Carolinae (2024): 53-62. <http://eudml.org/doc/299949>.

@article{ElMouadine2024,
abstract = {Let $L(H)$ denote the algebra of all bounded linear operators on a complex infinite dimensional Hilbert space $H$. For $A,B\in L(H)$, the generalized derivation $\delta _\{A,B\}$ and the multiplication operator $M_\{A,B\}$ are defined on $L(H)$ by $\delta _\{A,B\}(X)=AX-XB$ and $M_\{A,B\}(X)=AXB$. In this paper, we give a characterization of bounded operators $A$ and $B$ such that the range of $M_\{A,B\}$ is closed. We present some sufficient conditions for $\delta _\{A,B\}$ to have closed range. Some related results are also given.},
author = {El Mouadine, Hamza, Faouzi, Abdelkhalek, Bouhafsi, Youssef},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {generalized derivation; elementary operator; generalized inverse; Kato spectrum},
language = {eng},
number = {1},
pages = {53-62},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the range of some elementary operators},
url = {http://eudml.org/doc/299949},
year = {2024},
}

TY - JOUR
AU - El Mouadine, Hamza
AU - Faouzi, Abdelkhalek
AU - Bouhafsi, Youssef
TI - On the range of some elementary operators
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2024
PB - Charles University in Prague, Faculty of Mathematics and Physics
IS - 1
SP - 53
EP - 62
AB - Let $L(H)$ denote the algebra of all bounded linear operators on a complex infinite dimensional Hilbert space $H$. For $A,B\in L(H)$, the generalized derivation $\delta _{A,B}$ and the multiplication operator $M_{A,B}$ are defined on $L(H)$ by $\delta _{A,B}(X)=AX-XB$ and $M_{A,B}(X)=AXB$. In this paper, we give a characterization of bounded operators $A$ and $B$ such that the range of $M_{A,B}$ is closed. We present some sufficient conditions for $\delta _{A,B}$ to have closed range. Some related results are also given.
LA - eng
KW - generalized derivation; elementary operator; generalized inverse; Kato spectrum
UR - http://eudml.org/doc/299949
ER -

References

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  9. Laursen K. B., Neumann M. M., An Introduction to Local Spectral Theory, London Mathematical Society Monographs, New Series, 20, The Clarendon Press, Oxford University Press, New York, 2000. MR1747914
  10. Mbekhta M., Résolvant généralisé et théorie spectrale, J. Operator Theory 21 (1989), no. 1, 69–105 (French). MR1002122
  11. Rosenblum M., 10.1215/S0012-7094-56-02324-9, Duke Math. J. 23 (1956), 263–269. MR0079235DOI10.1215/S0012-7094-56-02324-9
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