Numerical ranges of weighted composition operators

The operator that takes the function f to ψ f ∘ φ is a weighted composition operator. We study numerical ranges of some classes of weighted composition operators on H 2 , the Hardy–Hilbert space of the unit disc. We consider the case where φ is a rotation of the unit disc and identify a class of convexoid operators. In the case of isometric weighted composition operators we give a complete classification of their numerical ranges. We also consider the inclusion of zero in the interior of the numerical range.