  The numerical range of a bounded linear operator on a complex Banach space
need not be convex unlike that on a Hilbert space. The aim of this paper is to
study operators $T$ on $ \ell^2_p $ for which the numerical range is convex. We
also obtain a nice relation between $V(T)$ and $ V(T^t)$ considering $ T \in
\mathbb{L} (\ell_p^2) $ and $ T^t \in \mathbb{L} (\ell_q^2) ,$ where $T^t$
denotes the transpose of $T$ and $p$ and $q$ are conjugate real numbers i.e., $
1 <p,q< \infty $ and $ \frac{1}{p}+\frac{1}{q}=1.$
