Glasgow Mathematical Journal
We denote the numerical range of the normal operator T by W(T). A characterization is given to the points in W(T) that lie on the boundary. The collection of such boundary points together with the interior of the the convex hull of the spectrum of T will then be the set W(T). Moreover, it is shown that such boundary points reveal a lot of information about the normal operator. For instance, such a boundary point always associates with an invariant (reducing) subspace of the normal operator. It follows that a normal operator acting on a separable Hilbert space cannot have a closed strictly convex set as its numerical range. Similar results are obtained for the Davis-Wielandt shell of it normal operator. One can deduce additional information of the normal operator by studying the boundary of its Davis-Wielandt shell. Further extension of the result to the joint numerical range of commuting operators is discussed.
Li, C. K., & Poon, Y. T. (2009). SPECTRUM, NUMERICAL RANGE AND Davis-Wielandt Shell Of A Normal Operator. Glasgow Mathematical Journal, 51(1), 91-100.