Geodesic
On the range-kernel orthogonality of elementary operators
Mathematica Bohemica, Tome 140 (2015) no. 3, pp. 261-269

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Let $L(H)$ denote the algebra of operators on a complex infinite dimensional Hilbert space $H$. For $A, B\in L(H)$, the generalized derivation $\delta _{A,B}$ and the elementary operator $\Delta _{A,B}$ are defined by $\delta _{A,B}(X)=AX-XB$ and $\Delta _{A,B}(X)=AXB-X$ for all $X\in L(H)$. In this paper, we exhibit pairs $(A,B)$ of operators such that the range-kernel orthogonality of $\delta _{A,B}$ holds for the usual operator norm. We generalize some recent results. We also establish some theorems on the orthogonality of the range and the kernel of $\Delta _{A,B}$ with respect to the wider class of unitarily invariant norms on $L(H)$.
Let $L(H)$ denote the algebra of operators on a complex infinite dimensional Hilbert space $H$. For $A, B\in L(H)$, the generalized derivation $\delta _{A,B}$ and the elementary operator $\Delta _{A,B}$ are defined by $\delta _{A,B}(X)=AX-XB$ and $\Delta _{A,B}(X)=AXB-X$ for all $X\in L(H)$. In this paper, we exhibit pairs $(A,B)$ of operators such that the range-kernel orthogonality of $\delta _{A,B}$ holds for the usual operator norm. We generalize some recent results. We also establish some theorems on the orthogonality of the range and the kernel of $\Delta _{A,B}$ with respect to the wider class of unitarily invariant norms on $L(H)$.
DOI : 10.21136/MB.2015.144393
Classification : 47A30, 47A63, 47B10, 47B15, 47B20, 47B47
Keywords: derivation; elementary operator; orthogonality; unitarily invariant norm; cyclic subnormal operator; Fuglede-Putnam property 
Bouali, Said; Bouhafsi, Youssef. On the range-kernel orthogonality of elementary operators. Mathematica Bohemica, Tome 140 (2015) no. 3, pp. 261-269. doi: 10.21136/MB.2015.144393
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[1] Anderson, J.: On normal derivations. Proc. Am. Math. Soc. 38 (1973), 135-140. | DOI | MR | Zbl

[2] Berger, C. A., Shaw, B. I.: Selfcommutators of multicyclic hyponormal operators are always trace class. Bull. Am. Math. Soc. 79 (1974), 1193-1199. | MR | Zbl

[3] Bouali, S., Bouhafsi, Y.: On the range kernel orthogonality and {$P$}-symmetric operators. Math. Inequal. Appl. 9 (2006), 511-519. | MR | Zbl

[4] Delai, M. B., Bouali, S., Cherki, S.: A remark on the orthogonality of the image to the kernel of a generalized derivation. Proc. Am. Math. Soc. 126 French (1998), 167-171. | MR

[5] Duggal, B. P.: A perturbed elementary operator and range-kernel orthogonality. Proc. Am. Math. Soc. 134 (2006), 1727-1734. | DOI | MR | Zbl

[6] Duggal, B. P.: A remark on normal derivations. Proc. Am. Math. Soc. 126 (1998), 2047-2052. | DOI | MR | Zbl

[7] Gohberg, I. C., Kreĭn, M. G.: Introduction to the Theory of Linear Nonselfadjoint Operators. Translations of Mathematical Monographs 18 American Mathematical Society, Providence (1969), translated from the Russian, Nauka, Moskva, 1965. | DOI | MR | Zbl

[8] Kittaneh, F.: Normal derivations in norm ideals. Proc. Am. Math. Soc. 123 (1995), 1779-1785. | DOI | MR | Zbl

[9] Tong, Y.: Kernels of generalized derivations. Acta Sci. Math. 54 (1990), 159-169. | MR | Zbl

[10] Turnšek, A.: Orthogonality in {$\scr C_p$} classes. Monatsh. Math. 132 (2001), 349-354. | DOI | MR

[11] Turnšek, A.: Elementary operators and orthogonality. Linear Algebra Appl. 317 (2000), 207-216. | MR | Zbl

[12] Yoshino, T.: Subnormal operator with a cyclic vector. Tôhoku Math. J. II. Ser. 21 (1969), 47-55. | DOI | MR | Zbl

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