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

on

\ell^2_p ℓ p 2

for which the numerical range is convex. We also obtain a nice relation between

V(T) V ( T )

and

V(T^t) V ( T t )

considering

T\in\mathbb{L}(\ell_p^2) T ∈ L ( ℓ p 2 )

and

T^t\in\mathbb{L}(\ell_q^2) T t ∈ L ( ℓ q 2 )

, where

T^t T t

denotes the transpose of

T T

and

p p

and

q q

are conjugate real numbers, i.e.,

1 <p,q< \infty 1 < p , q < ∞

and

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

.