The Spectral Scale and the k-Numerical Range

Original · EN

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 b₁ = (c+c*)/2 and b₂ = (c-c*)/2i, and write B for the the spectral scale of b₁, b₂ with respect to tau. 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 lambda is a singular point on the boundary of (Wₖ)(c) if and only if lambda is an isolated extreme point of (Wₖ)(c). In this case lambda = (n/k)tau(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 (Wₖ)(c) is the finite union of line segments and curved real analytic arcs.