On normal derivations

Abstract:

Let ${\Delta _T}$ be the derivation on $\mathfrak {B}(\mathcal {H})$ defined by ${\Delta _T}(X) = TX - XT(T,X \in \mathfrak {B}(\mathcal {H}))$. We prove that if $T$ is an isometry or a normal operator, then the range of ${\Delta _T}$ is orthogonal to the null space of ${\Delta _T}$. Also, we prove that if $T$ is normal with an infinite number of points in its spectrum then the closed linear span of the range and the null space of ${\Delta _T}$ is not all of $\mathfrak {B}(\mathcal {H})$.