The existence of a maximizing vector for the numerical range of a compact operator

Let $X$ be a complex Lebesgue space with a unique duality map $J$ from $X$ to $X^*$, the conjugate space of $X$. Let $A$ be a compact operator on $X$. This paper focuses on properties of $W(A)=\{J(x)(A(x)):\|x\|=1\}$ and $\Lambda(A)=\sup\{\RE\alpha:\alpha\in W(A)\}$. We adapt an example due to Halmos to show that for $X=l_p, 1<p<\infty$, there is a compact operator $A$ on $l_p$ with $W(A)$ the semi-open interval $[-1,0)$. So $\Lambda(A)$ is not attained as a maximum. A corollary of the main result in this paper is that if $X=l_p,1<p<\infty$, and $\Lambda(A)\ne 0$, then $\Lambda(A)$ is attained as a maximum.