On the numerical range of operators on some special Banach spaces

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 $\frac{1}{p}+\frac{1}{q}=1.$