**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