THE EXISTENCE OF A MAXIMIZING VECTOR FOR THE NUMERICAL RANGE OF A COMPACT 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 compact operator on X, This paper focuses on properties of W(A) = {J(x)(A(x) : parallel to x paral lel to = 1} and Lambda(A) = sup{Re alpha : alpha is an element of W (A )}. We adapt an example due to Halmos to show that for X = l(p): 1 < p < infinity, there is a compact operator A on l(p) with W(A) the semi- open interval [-1, 0). So Lambda(A) is not attained as a maximum. A co rollary of the main result in this paper is that if X = l(p), 1 < p < infinity, and Lambda(A) not equal 0, then Lambda(A) is attained as a m aximum.