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 P 2 = 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 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 relation between the minimal canonical angle between the range and the null space of P and 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.