Abstract
As is well known on the weighted Bergman spaces over the unit disk, compactness of differences of two composition operators is characterized by certain cancellation property of the inducing maps at every bad boundary point, which makes each composition operator in the difference fail to be compact. Recently, the second and third authors obtained a result implying that double difference cancellation is not possible for linear combinations of three composition operators. In this paper, we obtain a complete characterization for compact double differences formed by four composition operators. Applying our characterization, we easily recover known results on linear combinations of two or three composition operators. As another application, we also show that double difference cancellation is possible for linear combinations of four composition operators by constructing an explicit example of a compact double difference formed by two noncompact differences. In spite of such an example, our characterization also shows that double difference cancellation may occur in the global sense only, and that genuine double difference cancellation is not possible in a certain local sense.
| Original language | English |
|---|---|
| Journal | Journal of Functional Analysis |
| DOIs | |
| Publication status | Accepted/In press - 2016 May 24 |
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Keywords
- Compact operator
- Composition operator
- Double difference
- Primary
- Secondary
- Weighted Bergman space
ASJC Scopus subject areas
- Analysis
Cite this
Compact double differences of composition operators on the Bergman spaces. / Choe, Boo Rim; Koo, Hyung Woon; Wang, Maofa.
In: Journal of Functional Analysis, 24.05.2016.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Compact double differences of composition operators on the Bergman spaces
AU - Choe, Boo Rim
AU - Koo, Hyung Woon
AU - Wang, Maofa
PY - 2016/5/24
Y1 - 2016/5/24
N2 - As is well known on the weighted Bergman spaces over the unit disk, compactness of differences of two composition operators is characterized by certain cancellation property of the inducing maps at every bad boundary point, which makes each composition operator in the difference fail to be compact. Recently, the second and third authors obtained a result implying that double difference cancellation is not possible for linear combinations of three composition operators. In this paper, we obtain a complete characterization for compact double differences formed by four composition operators. Applying our characterization, we easily recover known results on linear combinations of two or three composition operators. As another application, we also show that double difference cancellation is possible for linear combinations of four composition operators by constructing an explicit example of a compact double difference formed by two noncompact differences. In spite of such an example, our characterization also shows that double difference cancellation may occur in the global sense only, and that genuine double difference cancellation is not possible in a certain local sense.
AB - As is well known on the weighted Bergman spaces over the unit disk, compactness of differences of two composition operators is characterized by certain cancellation property of the inducing maps at every bad boundary point, which makes each composition operator in the difference fail to be compact. Recently, the second and third authors obtained a result implying that double difference cancellation is not possible for linear combinations of three composition operators. In this paper, we obtain a complete characterization for compact double differences formed by four composition operators. Applying our characterization, we easily recover known results on linear combinations of two or three composition operators. As another application, we also show that double difference cancellation is possible for linear combinations of four composition operators by constructing an explicit example of a compact double difference formed by two noncompact differences. In spite of such an example, our characterization also shows that double difference cancellation may occur in the global sense only, and that genuine double difference cancellation is not possible in a certain local sense.
KW - Compact operator
KW - Composition operator
KW - Double difference
KW - Primary
KW - Secondary
KW - Weighted Bergman space
UR - http://www.scopus.com/inward/record.url?scp=84995960043&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84995960043&partnerID=8YFLogxK
U2 - 10.1016/j.jfa.2016.08.006
DO - 10.1016/j.jfa.2016.08.006
M3 - Article
AN - SCOPUS:84995960043
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
SN - 0022-1236
ER -