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

Abstract

In this paper we investigate the numerical range of C*bφm Caφn and Caφn C*bφm 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.

Keywords

• composition operator
• numerical range
• inner function
• weighted shift
• 47b33
• 47a12
• 15a60