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

Paul S. Bourdon and Joel H. Shapiro

Integral Equations and Operator Theory 44 (2002) 410--441

Abstract: We consider the numerical ranges of composition operators on the Hardy space H2 and, for inducing maps that fix a point of the unit disc, completely describe when zero is in the numerical range of the associated composition operator. We show that if zero is not in the numerical range then the operator is strictly positive definite, and in this case we determine when the numerical range lies in an acute sector of the right half-plane.In the course of our investigation we uncover surprising connections between composition operators, Chebyshev polynomials, and Pascal matrices.

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