Compact Differences of Composition Operators in Several Variables

Abstract

When ϕ and ψ are linear-fractional self-maps of the unit ball BN in C N , N ≥ 1, we show that the difference Cϕ − C ψ cannot be nontrivially compact on either the Hardy space H 2 (BN ) or any weighted Bergman space A 2 α (BN ). Our arguments emphasize geometrical properties of the inducing maps ϕ and ψ.

Compactness of composition operators on the Bergman spaces of convex domains and analytic discs

“…The authors of [AGT10] studied the closed range property of composition operators on the unit disc. Work on essential norm estimates and compactness of composition operators was studied on the ball in C n and on the unit disc in C by [CM95] and [HMW11]. On more general bounded strongly pseudoconvex domains in C n , [ ČZ07] studied the essential norm of the composition operator in terms of the behavior of the norm of the normalized Bergman kernel composed with the symbol.…”

Compactness and Singular Points of Composition Operators on Bergman spaces

“…The authors of [AGT10] studied the closed range property of composition operators on the unit disk. Work on essential norm estimates and compactness of composition operators was studied on the ball in C n and on the unit disk in C by [CM95] and [HMW11]. On more general bounded strongly pseudoconvex domains in C n , [ ČZ07] studied the essential norm of the composition operator in terms of the behavior of the norm of the normalized Bergman kernel composed with the symbol.…”

Differences of weighted differentiation composition operators from α-Bloch space to H∞ space

“…Soon after Song and Zhou [16] improved such characterizations for the high dimensional cases. For further references and details about the difference of two (weighted) composition operators, see [2,3,5,6,9,14,17,18].…”