On the field of values of oblique projections

Abstract

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 (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.