When is zero in the numerical range of a composition operator?

We work on the Hardy space H 2 of the open unit disc U and consider the numerical ranges of composition operators C (phi) induced by holomorphic self-maps (phi) of U. For maps (phi) that fix a point of U we determine precisely when 0 belongs to the numerical range W of C(phi), and in the process discover the following dichotomy: either 0 (element of) W or the real part of C (phi) admits a decomposition that reveals it to bestrictly positive-definite. In this latter case we characterize those operators that aresectorial. For compact composition operators our work has the following consequences: it yields a complete description of the corner points of the closure of W, and it establishes when W is closed. In the course of our investigation we uncover surprising connections between composition operators, Chebyshev polynomials, and Pascal matrices.