The compactness of the sum of weighted composition operators on the ball algebra

Ce-Zhong Tong,Ze-Hua Zhou

doi:10.1186/1029-242x-2011-45

Abstract

In this paper, we investigate the compactness of the sum of weighted composition operators on the unit ball algebra, and give the characterization of compact differences of two weighted composition operators on the ball algebra. The connectness of the topological space consisting of non-zero weighted composition operators on the unit ball algebra is also studied. 2000 Mathematics Subject Classification. Primary: 47B33; Secondary: 47B38, 46E15, 32A36.

Highlights

Let H(BN) be the class of all holomorphic functions on BN and S(BN ), the collection of all holomorphic self mappings of BN, where BN is the unit ball in the N-dimensional complex space CN
Denote by A = A(BN), the unit ball algebra of all continuous functions on BN that are holomorphic on BN
Hosokawa and co-workers [13] studied the topological components of the topological space of weighted composition operators on the space of bounded analytic functions on the open unit disk in the uniform operator topology

Summary

Introduction

Let H(BN) be the class of all holomorphic functions on BN and S(BN ), the collection of all holomorphic self mappings of BN, where BN is the unit ball in the N-dimensional complex space CN. Hosokawa and co-workers [13] studied the topological components of the topological space of weighted composition operators on the space of bounded analytic functions on the open unit disk in the uniform operator topology These results were extended to the setting of H∞(BN) by Toews [14], and independently by Gorkin and co-workers [15], and the setting of H∞(DN) by Fang and Zhou [12], where DN is the unit polydisk. From the item (3), we can choose the subsequences ( denoted by {wn}) satisfying (5): φs(wn)(φl(wn)), φs(wn) → σls = 0 For such a sequence {wn}, define the functions fn(z) =( φs(wn)(z), φs(wn) − 1)( φs(wn)(z), φs(wn) − λ).

Now define a function G on BN by setting

Hence p