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Generalized D-symmetric Operators II

  • S. Bouali (a1) and M. Ech-chad (a2)

Abstract

Let $H$ be a separable, infinite-dimensional, complex Hilbert space and let $A,\,B\,\in \,\mathcal{L}\left( H \right)$ , where $\mathcal{L}(H)$ is the algebra of all bounded linear operators on $H$ . Let ${{\delta }_{AB}}\,:\mathcal{L}\left( H \right)\to \mathcal{L}\left( H \right)$ denote the generalized derivation ${{\delta }_{AB}}\left( X \right)\,=\,AX\,-\,XB$ . This note will initiate a study on the class of pairs $\left( A,\,B \right)$ such that $\overline{R\left( {{\delta }_{AB}} \right)}\,=\,\overline{R\left( {{\delta }_{A*\,B*}} \right)}$ .

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References

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[1] Anderson, J., Bunce, J. W., Deddens, J. A., and Williams, J. P., C*-algebras and derivation ranges. Acta Sci. Math. (Szeged) 40(1978), no. 3–4, 211227.
[2] Anderson, J. and Foias, C., Properties which normal operators share with normal derivation and related operators. Pacific J. Math. 61(1975), no. 2, 313325.
[3] Benlarbi, M., Bouali, S., and Cherki, S., Une remarque sur l’orthogonalité de l’image au noyau d’une dérivation généralisée. Proc. Amer. Math. Soc. 126(1998), no. 1, 167171. doi:10.1090/S0002-9939-98-03996-3
[4] Bouali, S. and Charles, J., Extension de la notion d’opérateur D-symétrique. I. Acta Sci. Math. (Szeged) 58(1993), no. 1–4, 517525.
[5] Bouali, S. and Charles, J., Extension de la notion d’opérateur D-symétrique. II. Linear Algebra Appl. 225(1995), 175185. doi:10.1016/0024-3795(94)00003-V
[6] Herrero, D. A., Approximation of Hilbert space operators. Vol. I, Research Notes in Mathematics, 72, Pitman (Advanced Publishing Program), Boston, MA, 1982.
[7] Rosenblum, M., On the operator equation BX – XA = Q. Duke Math. J. 23(1956), 263269. doi:10.1215/S0012-7094-56-02324-9
[8] Stampfli, J. G., On self-adjoint derivation ranges. Pacific J. Math. 82(1979), no. 1, 257277.
[9] Williams, J. P., Derivation ranges: open problems. In: Topics in modern operator theory, Operator Theory: Adv. Appl., 2, Birkhäuser, Basel-Boston, MA, 1981, pp. 319328.
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Generalized D-symmetric Operators II

  • S. Bouali (a1) and M. Ech-chad (a2)

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