AN EIGENVALUE PROBLEM FOR THE NUMERICAL RANGE OF A BOUNDED LINEAR OPERATOR

Let X be a complex Lebesgue space with a unique duality map J from X t o X, the conjugate space of X. Let A be a bounded linear operator on X. In this paper we obtain a non-linear eigenvalue problem for Lambda( A) = sup{Re alpha : alpha is an element of W(A)} where W(A) = {J(x)A(x )) : parallel to x parallel to =1}, under the assumption that Lambda(A ) is attained as a maximum. We then use the results to compute Lambda( A) and the convex hull of W(A) for some linear operators A on l(p), 2 < p < infinity.