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 l(p)(2) for which the numerical range is convex. We also obtain a nice relation between V(T) and V(T-t) considering T is an element of L(l(p)(2) ) and T-t is an element of L(l(q)(2)), where T-t denotes the transpose of T and p and q are conjugate real numbers i.e., 1 < p, q < infinity and 1/p + 1/q = 1.