On compactness of the difference of composition operators

Abstract

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