John Clifford, Michael Dabkowski, Alan Wiggins
March 11, 2021
In this paper we investigate the numerical range of C * bφ m C aφ n and C aφ 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.