Not every $d$-symmetric operator is GCR

Abstract:

Let $T$ be an element of $\mathcal {B}(\mathcal {H})$, the algebra of bounded linear operators on the Hilbert space $\mathcal {H}$. The derivation induced by $T$ is the map ${\delta _T}(X) = TX - XT$ from $\mathcal {B}(\mathcal {H})$ into itself. $T$ is $d$-symmetric if the norm closure of the range of ${\delta _T}$, $\mathcal {R}{({\delta _T})^\_}$, is closed under taking adjoints. This paper answers the question of whether every $d$-symmetric operator is GCR by giving an example of an NGCR weighted shift that is also $d$-symmetric.