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

Анотація

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 $\delta_{A,B}$.

Біографії авторів

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

Посилання

J.H. Anderson, On normal derivations, Proc. Amer. Math. Soc., 38 (1973), 135–140.

J.H. Anderson, C. Fioas, Properties which normal operators share with normal derivations and related operators, Pacific J. Math., 61 (1975), 313–325.

C.A. Berger, B.I. Shaw, Self-commutators of multicyclic hyponormal operators are always trace class, Bull. Amer. Math. Soc., 79 (1973), 1193–1199.

S. Bouali, J. Charles, Extension de la notion d’op´erateur-sym´etrique I, Acta. Sci. Math. (Szeged), 58 (1993), 517–525.

S. Bouali, S. Cherki, Approximation by generalized commutators, Acta Sci. Math (Szeged), 63 (1997), 273–278.

M. Benlarbi Delai, S. Bouali, S. Cherki, Une remarque sur l’orthogonalit´e de l’image au noyau d’une d´erivation g´en´eralis´ee, Proc. Amer. Math. Soc., 126 (1998), 167–171.

S. Bouali, Y. Bouhafsi, On the range-kernel orthogonality and P-symmetric operators, Math. Inequal. Appl., 9 (2006), 511–519.

S. Bouali, Y. Bouhafsi, P-symmetric operators and the range of a subnormal derivation, Acta Sci. Math. (Szeged), 72 (2006), 701–708.

S. Bouali, M. Ech-chad, Analytic fonctions, derivations and orthogonality, preprint.

B.P. Duggal, A remark on normal derivations, Proc. Amer. Math. Soc., 126 (1998), 2047–2052.

B.P. Duggal, On the range-kernel orthogonality of derivations, Linear Algebra Appl., 304 (2000), 103–108.

B.P. Duggal, Putnam-Fuglede theorem and the range-kernel orthogonality of derivations, International J. Math. and Math. Sciences, 27 (2001), 573–582.

B.P. Duggal, A perturbed elementary operator and range-kernel orthogonality, Proc. Amer. Math. Soc., 134 (2006), 1727–1734.

P.A. Fillmore, J.G. Stampfli, J.P. Williams, On the essential numerical range, the essential spectrum, and a problem of Halmos, Acta Sci. Math., 33 (1972), 179–192.

T. Furuta, Relaxation of normality in the Fuglede-Putnam theorem, Proc. Amer. Math. Soc., 77 (1979), 324–328.

F. Kittaneh, Operators that are orthogonal to the range of a derivation, J. Math. Anal. Appl., 203 (1997), 868–873.

R.C. James, Orthogonality and linear functionals in normed linear spaces, Trans. Amer. Math. Soc., 61 (1947), 265–292.

P. Rosenthal, On the equations X = KXS and AX = XK, Spectral theory, 8 (1982), 389–391.

J.G. Stampfli, B.L. Wadhwa, On dominant operators, Monatshefte f¨ur Mathematik, 84 (1977), 143–153.

Y. Tong, Kernels of generalized derivations, Acta. Sci. Math(Szeged), 54 (1990), 159–169.

A. Turnsek, Orthogonality in Cp classes, Monatsh. Math. 132 (2001), 349–354.

J.P. Williams, Finite operators, Proc. Amer. Math. Soc., 26 (1970), 129–136.

T. Yoshino, Subnormal operators with a cyclic vector, Tˆohoku Math. J., 21 (1969), 47–55.

Опубліковано
2022-06-27
Як цитувати
Bouhafsi, Y., Ech-chad, M., Missouri, M., & Zouaki, A. (2022). Remarks on the range and the kernel of generalized derivation. Математичні студії, 57(2), 202-209. https://doi.org/10.30970/ms.57.2.202-209
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