Rocky Mountain Journal of Mathematics
Operators on function paces acting by composition to the right with a fixed self-map φ of some set are called composition operators with the symbol φ. In this paper, composition operators on the Hilbert Hardy space over the unit disk are considered. The numerical ranges of composition operators with inner symbol of parabolic automorphic type of hyperbolic type are shown to be circular.
Matache, Valentin, "Numerical Ranges of Composition Operators with Inner Symbols" (2012). Mathematics Faculty Publications. 32.