Schatten class composition operators

Let Cφ be a composition operator on the Bergman space A² of the unit disc. A well-known problem asks whether the condition _∫D (Equation present) ^p dλ(z) < ∞ is equivalent to the membership of Cφ in the Schatten class Cp, 1 < p < ∞. This was settled in the negative for the case 2 < p < ∞ in [3]. When 2 < p < ∞, this condition is not sufficient for Cφ ∈ Cp. In this article, we take up the case 1 < p < 2. We show that when 1 < p < 2, this condition is not necessary for Cφ ∈ Cp.