Class A composition operators on H2

In this paper, we study class A   composition operators CφCφ on the Hardy space H2H2. We show that if CφCφ belongs to class A, then 0 is a fixed point of the symbol φ. As a corollary, we obtain that every invertible class A composition operator is unitary. Moreover, we examine spectral properties and the commutants of class A composition operators. We also prove that if φ   is a linear fractional self-map of DD into itself, then CφCφ belongs to class A   if and only if it is subnormal. Finally, we provide some conditions under which Cφ⁎ belongs to class A.