AbstractFor a continuous linear operator A on a Hilbert space X and unit vectors x and y, an investigation of the set W[x,y]={z∗Az:z∗z=1 and zϵspan{x,y}} reveals several new results about W(A), the numerical range of A. W[x,y] is an elliptical disk (possibly degenerate), and several conditions are given which imply that W[x,y] is a line segment. In particular if x is a reducing eigenvector of A, then W[x,y] is a line segment. A unit vector is called interior (boundary) if x∗Ax is in the interior (boundary) of W(A). It is shown that interior reducing eigenvectorsare orthogonal to all boundary vectors and that boundary eigenvectors are orthogonal to all other boundary vectors y [except possibly when y∗ Ay is interior to a line segment in the boundary of W(A) through the given eigenvalue].