Operators with closed numerical ranges in nest algebras

Abstract:

In the present paper, we continue our research on numerical ranges of operators. With newly developed techniques, we show that

Let $\mathcal {N}$ be a nest on a Hilbert space $\mathcal {H}$ and $T\in \mathcal {T} (\mathcal {N})$, where $\mathcal {T} (\mathcal {N})$ denotes the nest algebra associated with $\mathcal {N}$. Then for given $\varepsilon >0$, there exists a compact operator $K$ with $\|K\|<\varepsilon$ such that $T+K \in \mathcal {T} (\mathcal {N})$ and the numerical range of $T+K$ is closed.

As applications, we show that the statement of the above type holds for the class of Cowen-Douglas operators, the class of nilpotent operators and the class of quasinilpotent operators.