Salah Mecheri

Generalized Derivation and Double Operator Integrals

Abstract:
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 $\delta_{A,B}:\mathbb{B}(H)\mapsto \mathbb{B}(H)$ by $\delta_ {A,B}(X)=AX - XB$. In this paper we consider the question posed by Turnsek in 2003, when $\overline{\ran (\delta_{A,B}\mid _{C_{p}})}^{c_{p}}=\overline{\ran (\delta_{A,B}\cap_{C_{p}})}^{c_{p}}?$ 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 2000: Primary: 47B47, 47B10, 47B21, 47B49. Secondary: 47A30, 47A13, 47A60