  In this work we study the essential spectra of composition operators on Hardy
spaces of analytic functions which might be termed as "quasi-parabolic". This
is the class of composition operators on H^{2} with symbols whose conjugate
with the Cayley transform on the upper half-plane are of the form \phi(z) =
z+\psi(z) where \psi\in H^{2}(\mathbb{H}) and \Im(\psi(z)) >\delta > 0. We
especially examine the case where \psi is discontinuous at infinity. A new
method is devised to show that this type of composition operators fall in a
C*-algebra of Toeplitz operators and Fourier multipliers. This method enables
us to provide new examples of essentially normal composition operators and to
calculate their essential spectra.
