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 τ denote the normalized trace on B(H). Set b1 = (c + c )/2 and b2 = (c − c)/2i, and write B for the the spectral scale of {b1, b2} with respect to τ . We show that B contains full information about Wk(c), the k-numerical range of c for each k = 1, . . . , n. We then use our previous work on spectral scales to prove several new facts about Wk(c). For example, we show in Theorem 3.4 that the point λ is a singular point on the boundary of Wk(c) if and only if λ is an isolated extreme point of Wk(c). In this case λ = (n/k)τ(cz), where z is a central projection in in the algebra generated by b1, b2 and the identity. We show in Theorem 3.5, that c is normal if and only if Wk(c) is a polygon for each k. Finally, it is shown in Theorem 5.4 that the boundary of the k-numerical range is the finite union of line segments and curved real analytic arcs. 0 Introduction and Notation The spectral scale was introduced by the present authors and Nik Weaver in [2] and further developed by the authors in [3]. It is defined for any finite set of self-adjoint operators in a finite von Neumann algebra. The main theme in [2] and [3] is that full spectral information about real linear combinations ∗The second author was partially supported by the National Science Foundation during the period of research that resulted in this paper.