On operators with closed numerical ranges

Abstract

In this article we investigate the numerical ranges of several classes of operators. It is shown that, if we let $T$ be a hyponormal operator and let $\epsilon >0$, then there exists a compact operator $K$ with norm less than $\epsilon$ such that $T+K$ is hyponormal and has a closed numerical range. Moreover we prove that the statement of the above type holds for other operator classes, including weighted shifts, normaloid operators, triangular operators, and block-diagonal operators.