Eigenvalues in the boundary of the numerical range

Abstract

We study eigenvalues of a continuous linear operator T on a complex Banach space X that lie in the boundary of the numerical range of T. We show that the kernel of Tl is orthogonal, in the sense of G. Birkhoff, to the range of T-l. M. R. Fortet [5, Th. Ill, p. 32] proves that if T is a continuous linear operator of norm one on a strictly convex Banach space, then the kernel of T -I is orthogonal to the range of T -I. Proposition 1 is a generalisation of this result, since the numerical radius is less than or equal to the norm [2, Th. 4.1]. Proposition 1 is also related to the theorem of N. Nirschl and H. Schneider that an eigenvalue in the boundary of the numerical range has ascent one