On compactness of the difference of composition operators ✩

Let φ and ψ be analytic self-maps of the unit disc, and denote by Cφ and Cψ the induced composition operators. The compactness and weak compactness of the difference T = Cφ −Cψ are studied on Hp spaces of the unit disc and Lp spaces of the unit circle. It is shown that the compactness of T on Hp is independent of p ∈ [1,∞). The compactness of T on L1 and M (the space of complex measures) is characterized, and examples of φ and ψ are constructed such that T is compact on H1 but non-compact on L1. Other given results deal with L∞, weakly compact counterparts of the previous results, and a conjecture of J.E. Shapiro.  2004 Elsevier Inc. All rights reserved.