On Generalized Numerical Ranges of Quadratic Operators
  It is shown that the result of Tso-Wu on the elliptical shape of the numerical range of quadratic operators holds also for the essential numerical range. The latter is described quantitatively, and based on that sufficient conditions are established under which the c-numerical range also is an ellipse. Several examples are considered, including singular integral operators with the Cauchy kernel and composition operators. 
Leiba Rodman, Ilya M. Spitkovsky