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 ${\mathrm{\ell }}_p^2$ 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$denotesthetransposeof$T$and$p$and$q$areconjugaterealnumbersi\mathrm{.}e\mathrm{.},$ 1 <p,q< \infty $and$ \frac{1}{p}+\frac{1}{q}=1.$