The Numerical Range of C* ψ Cφ and Cφ C* ψ

Abstract

Abstract In this paper we investigate the numerical range of C* bφ m Caφ n and Caφ n C* bφ m on the Hardy space where φ is an inner function fixing the origin and a and b are points in the open unit disc. In the case when |a| = |b| = 1 we characterize the numerical range of these operators by constructing lacunary polynomials of unit norm whose image under the quadratic form incrementally foliate the numerical range. In the case when a and b are small we show numerical range of both operators is equal to the numerical range of the operator restricted to a 3-dimensional subspace.