On the field of values of oblique projections

We highlight some properties of the field of values (or numerical range) W(P) of an oblique projector P on a Hilbert space, i.e., of an operator satisfying P2=P. If P is neither null nor the identity, we present a direct proof showing that W(P)=W(I-P), i.e., the field of values of an oblique projection coincides with that of its complementary projection. We also show that W(P) is an elliptical disk (i.e., the set of points circumscribed by an ellipse) with foci at 0 and 1 and eccentricity 1/‖P‖. These two results combined provide a new proof of the identity ‖P‖=‖I-P‖. We discuss the influence of the minimal canonical angle between the range and the null space of P, on the shape of W(P). In the finite dimensional case, we show a relation between the eigenvalues of matrices related to these complementary projections and present a second proof to the fact that W(P) is an elliptical disk.