Georgian Mathematical Journal 12 (2005), No. 4, 717--726
Copyright Heldermann Verlag 2005
Generalized Derivation and Double Operator Integrals
Salah Mecheri
Dept. of Mathematics, King Saud University, College of Science, P. O. Box 2455, Riyadh 11451, Saudi Arabia
mecherisalah@hotmail.com
[Abstract-pdf]
Let $H$ be a separable infinite dimensional complex Hilbert space, and let $\mathbb{B}(H)$ denote the algebra of all bounded linear operators on $H$. Let $A, B$ be operators in $\mathbb{B}(H)$. We define the generalized derivation $\delta_{A,B} \colon \mathbb{B}(H) \mapsto \mathbb{B}(H)$ by $\delta_ {A,B}(X) = AX - XB$. In this paper we consider the question posed by Turnsek in 2003, when $\overline{\text{ran} (\delta_{A,B} \mid_{C_{p}})}^{c_{p}} = \overline{\text{ran} (\delta_{A,B}\cap_{C_{p}})}^{c_{p}}$? We prove that this holds in the case where $A$ and $B$ satisfy the Fuglede-Putnam theorem. Finally, we apply the obtained results to double operator integrals.
Keywords: Generalized derivation, Fuglede-Putnam theorem, Hilbert-Schmidt class, double operator integrals.
MSC: 47B47, 47B10, 47B21, 47B49; 47A30, 47A13, 47A60
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