THE SPECTRAL SCALE AND THE k–NUMERICAL RANGE

Suppose that c is a linear operator acting on an n -dimensional complex Hilbert Space H , and let (tau) denote the normalized trace on B(H) . Set b_1 = (c+c^*)/2 and b_2 = (c-c^*)/2i , and write B for the spectral scale of {b_1, b_2} with respect to (tau) . We show that B contains full information about W_k(c) , the k -numerical range of c for each k = 1,...,n . This is in addition to the matrix pencil information that has been described in previous papers. Thus both types of information are contained in the geometry of a single 3-dimensional compact, convex set. We then use spectral scales to prove a new fact about W_k(c) . We show in Theorem 3.4 that the point \lambda is a singular point on the boundary of W_k(c) if and only if (lambda) is an isolated extreme point of W_k(c) : i.e. it is the end point of two line segments on the boundary of W_k(c) . In this case (lambda) = (n/k)(tau)(cz) , where z is a central projection in the algebra generated by c and the identity. In addition we show how the general theory of the spectral scale may be used to derive some other known properties of the knumerical range.