On the field of values of oblique projections (Q979015)

The authors consider properties of the numerical ranges \(W(P)\) of an oblique projection (or an idempotent operator) \(P\), that is, a bounded linear operator \(P\) on a complex Hilbert space satisfying \(P^2= P\). They show that, if \(P\) is idempotent, nonzero and not the identity operator, then \(W(P)= W(I- P)\). Then they give a direct proof of the known fact that the numerical range of an idempotent \(P\) is the (open or closed) elliptic disc (possibly degenerate) with foci \(0\) and \(1\) and minor axis of length \((\| P\|^2- 1)^{1/2}\). It follows in particular that \(W(P^*)= W(P)\). In the finite-dimensional case, they present a second proof of the fact that \(W(P)\) is an elliptic disc. We remark that all the results in this paper are either known or easily deducible from the canonical model for the more general quadratic operators as presented in [\textit{S.-H.\thinspace Tso} and the reviewer, Rocky Mt.\ J.\ Math.\ 29, No.\,3, 1139--1152 (1999Zbl 0957.47005)].