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

We work on the Hardy spaceH2 of the open unit disc $$\mathbb{U}$$ and consider the numerical ranges of composition operatorsCf induced by holomorphic self-maps f of $$\mathbb{U}$$. For maps f that fix a point of $$\mathbb{U}$$ we determine precisely when 0 belongs to the numerical rangeW ofCf, and in the process discover the following dichotomy: either 0?W or the real part ofCf 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 ofW, and it establishes whenW is closed. In the course of our investigation we uncover surprising connections between composition operators, Chebyshev polynomials, and Pascal matrices.