Compact and Weakly Compact Composition Operators on BMOA
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- 作者：Jussi Laitila (1)
Pekka J. Nieminen (2)
Eero Saksman (2)
Hans-Olav Tylli (2) - 关键词：Primary 47B33 ; Secondary 30D50 ; 46E15 ; 47B07
- 刊名：Complex Analysis and Operator Theory
- 出版年：2013
- 出版时间：February 2013
- 年：2013
- 卷：7
- 期：1
- 页码：163-181
- 全文大小：285KB
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Pekka J. Nieminen (2)
Eero Saksman (2)
Hans-Olav Tylli (2)
1. ISER, University of Essex, Colchester, CO4 3SQ, UK
2. Department of Mathematics and Statistics, University of Helsinki, PO Box 68, 00014, Helsinki, Finland - ISSN：1661-8262
文摘
Any analytic map φ of the unit disc ${\mathbb{D}}$ into itself induces a composition operator C φ on BMOA, mapping ${f \mapsto f \circ \varphi}$ , where BMOA is the Banach space of analytic functions ${f\colon \mathbb{D} \to \mathbb{C}}$ whose boundary values have bounded mean oscillation on the unit circle. We show that C φ is weakly compact on BMOA precisely when it is compact on BMOA, thus solving a question initially posed by Tjani and by Bourdon, Cima and Matheson in the special case of VMOA. As a crucial step of our argument we simplify the compactness criterion due to Smith for C φ on BMOA and show that his condition on the Nevanlinna counting function alone characterizes compactness. Additional equivalent compactness criteria are established. Furthermore, we prove the unexpected result that compactness of C φ on VMOA implies compactness even from the Bloch space into VMOA.