Abstract
For 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 zϵ span{ 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].
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