Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
generalized derivation; elementary operator; orthogonality; subnormal operator; compact operator
generalized derivation; elementary operator; orthogonality; subnormal operator; compact operator
Summary:
Let $L(H)$ denote the algebra of all bounded linear operators on a complex infinite dimensional Hilbert space $H$ and let $(\mathcal {J},\|{\cdot }\|_{\mathcal {J}})$ denote a norm ideal in $L(H)$. 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 $M_{A,B}(X)=AXB$. Denote $\tau _{A,B}=M_{A,B}-I$ and $d_{A,B}=\delta _{A,B}$ or $\tau _{A,B}$. We give some pairs $(A,B)$ of operators $A$ and $B$ such that the range $R(d_{A,B}|\mathcal {J})$ is orthogonal to the kernel $\ker (d_{A,B}|\mathcal {J})$ of the restriction $d_{A,B}|\mathcal {J}$ with respect to unitarily invariant norm $\|{\cdot }\|_{\mathcal {J}}$. It is proved that if $A$ and $B^{\ast }$ are subnormal, then the Hilbert-Schmidt operators in the range of $M_{A,B}$ are orthogonal to those in the kernel of $M_{A,B}$. We also give some related results.
References:
[1] Anderson, J.: On normal derivations. Proc. Am. Math. Soc. 38 (1973), 135-140. DOI 10.1090/S0002-9939-1973-0312313-6 | MR 0312313 | Zbl 0255.47036
[2] Anderson, J., Foiaş, C.: Properties which normal operators share with normal derivations and related operators. Pac. J. Math. 61 (1975), 313-325. DOI 10.2140/pjm.1975.61.313 | MR 0412889 | Zbl 0324.47018
[3] Delai, M. Benlarbi, Bouali, S., Cherki, S.: Une remarque sur l'orthogonalité de l'image au noyau d'une dérivation généralisée. Proc. Am. Math. Soc. 126 (1998), 167-171 French. DOI 10.1090/S0002-9939-98-03996-3 | MR 1416081 | Zbl 0889.47021
[4] Berger, C. A., Shaw, B. I.: Selfcommutators of multicyclic hyponormal operators are always trace class. Bull. Am. Math. Soc. 79 (1973), 1193-1199. DOI 10.1090/S0002-9904-1973-13375-0 | MR 0374972 | Zbl 0283.47018
[5] Bouali, S., Bouhafsi, Y.: On the range kernel orthogonality and $p$-symmetric operators. Math. Inequal. Appl. 9 (2006), 511-519. DOI 10.7153/mia-09-49 | MR 2242781 | Zbl 1112.47026
[6] Bouali, S., Cherki, S.: Approximation by generalized commutators. Acta Sci. Math. 63 (1997), 273-278. MR 1459791 | Zbl 0892.47035
[7] Duggal, B. P.: Range kernel orthogonality of derivations. Linear Algebra Appl. 304 (2000), 103-108. DOI 10.1016/S0024-3795(99)00193-7 | MR 1734207 | Zbl 0955.47008
[8] Duggal, B. P.: A perturbed elementary operator and range-kernel orthogonality. Proc. Am. Math. Soc. 134 (2006), 1727-1734. DOI 10.1090/S0002-9939-05-08337-1 | MR 2204285 | Zbl 1082.47031
[9] Dunford, N., Schwartz, J. T.: Linear Operators. Part I. General Theory. Pure and Applied Mathematics 7. Interscience, New York (1958). MR 0117523 | Zbl 0084.10402
[10] Herrero, D. A.: Approximation of Hilbert Space Operator. I. Research Notes in Mathematics 72. Pitman, Boston (1982). MR 0676127 | Zbl 0494.47001
[11] Kittaneh, F.: Normal derivations in norm ideals. Proc. Am. Math. Soc. 123 (1995), 1779-1785. DOI 10.1090/S0002-9939-1995-1242091-2 | MR 1242091 | Zbl 0831.47036
[12] Kittaneh, F.: Operators that are orthogonal to the range of a derivation. J. Math. Anal. Appl. 203 (1997), 868-873. DOI 10.1006/jmaa.1996.0418 | MR 1417136 | Zbl 0869.47015
[13] Mazouz, A.: On the range and the kernel of the operator $X\longmapsto AXB-X$. Proc. Am. Math. Soc. 127 (1999), 2105-2107. DOI 10.1090/S0002-9939-99-04754-1 | MR 1487327 | Zbl 0922.47030
[14] Morjane, M., Azhoum, K., Ech-Chad, M., Bouhafsi, Y.: On the Fuglede-Putnam-Aluthge property. J. Anal. 33 (2025), 2047-2060. DOI 10.1007/s41478-025-00903-y | MR 4975796
[15] Morjane, M., Madani, S., Ech-Chad, M., Bouhafsi, Y.: Putnam-Fuglede theorems and orthogonality of elementary operators in $\mathcal{C}_{p}$ classes. Proc. Inst. Math. Mech., Natl. Acad. Sci. Azerb. 50 (2024), 274-284. DOI 10.30546/2409-4994.2024.50.2.274 | MR 4838754 | Zbl 07988414
[16] Turnšek, A.: Elementary operators and orthogonality. Linear Algebra Appl. 317 (2000), 207-216. DOI 10.1016/S0024-3795(00)00163-4 | MR 1782211 | Zbl 1084.47510
[17] Turnšek, A.: Orthogonality in $C_p$ classes. Monatsh. Math. 132 (2001), 349-354. DOI 10.1007/s006050170039 | MR 1844072 | Zbl 0992.47010
[18] Yoshino, T.: Subnormal operators with a cyclic vector. Tohoku Math. J., II. Ser. 21 (1969), 47-55. DOI 10.2748/tmj/1178243033 | MR 0246145 | Zbl 0192.47801
Search
Partner of
