Cancellation properties of composition operators on Bergman spaces

Hyung Woon Koo, Maofa Wang

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

The compact difference of two composition operators on the Bergman spaces over the unit disc is characterized in [11] in terms of certain cancellation property of the inducing maps at every "bad" boundary points, which make each single composition operator not to be compact. In this paper, we completely characterize the compactness of a linear combination of three composition operators on the Bergman space. As one consequence of this characterization, we show that there is no cancellation property for the compactness of double difference of composition operators. More precisely, we show that if ϕ<inf>i</inf> are distinct and none of Cϕi is compact, then (C<inf>ϕ1-Cϕ2</inf>)-(C<inf>ϕ3-Cϕ1</inf>) is compact if and only if both (C<inf>ϕ1-Cϕ2</inf>) and (C<inf>ϕ3-Cϕ1</inf>) are compact.

Original languageEnglish
Pages (from-to)1174-1182
Number of pages9
JournalJournal of Mathematical Analysis and Applications
Volume432
Issue number2
DOIs
Publication statusPublished - 2015 Dec 15

Fingerprint

Bergman Space
Composition Operator
Cancellation
Chemical analysis
Compactness
Unit Disk
Linear Combination
If and only if
Distinct

Keywords

  • Compactness
  • Difference of composition operators
  • Linear combination

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

Cancellation properties of composition operators on Bergman spaces. / Koo, Hyung Woon; Wang, Maofa.

In: Journal of Mathematical Analysis and Applications, Vol. 432, No. 2, 15.12.2015, p. 1174-1182.

Research output: Contribution to journalArticle

@article{a25b01a030d448909f5c14f375d32f25,
title = "Cancellation properties of composition operators on Bergman spaces",
abstract = "The compact difference of two composition operators on the Bergman spaces over the unit disc is characterized in [11] in terms of certain cancellation property of the inducing maps at every {"}bad{"} boundary points, which make each single composition operator not to be compact. In this paper, we completely characterize the compactness of a linear combination of three composition operators on the Bergman space. As one consequence of this characterization, we show that there is no cancellation property for the compactness of double difference of composition operators. More precisely, we show that if ϕi are distinct and none of Cϕi is compact, then (Cϕ1-Cϕ2)-(Cϕ3-Cϕ1) is compact if and only if both (Cϕ1-Cϕ2) and (Cϕ3-Cϕ1) are compact.",
keywords = "Compactness, Difference of composition operators, Linear combination",
author = "Koo, {Hyung Woon} and Maofa Wang",
year = "2015",
month = "12",
day = "15",
doi = "10.1016/j.jmaa.2015.07.027",
language = "English",
volume = "432",
pages = "1174--1182",
journal = "Journal of Mathematical Analysis and Applications",
issn = "0022-247X",
publisher = "Academic Press Inc.",
number = "2",

}

TY - JOUR

T1 - Cancellation properties of composition operators on Bergman spaces

AU - Koo, Hyung Woon

AU - Wang, Maofa

PY - 2015/12/15

Y1 - 2015/12/15

N2 - The compact difference of two composition operators on the Bergman spaces over the unit disc is characterized in [11] in terms of certain cancellation property of the inducing maps at every "bad" boundary points, which make each single composition operator not to be compact. In this paper, we completely characterize the compactness of a linear combination of three composition operators on the Bergman space. As one consequence of this characterization, we show that there is no cancellation property for the compactness of double difference of composition operators. More precisely, we show that if ϕi are distinct and none of Cϕi is compact, then (Cϕ1-Cϕ2)-(Cϕ3-Cϕ1) is compact if and only if both (Cϕ1-Cϕ2) and (Cϕ3-Cϕ1) are compact.

AB - The compact difference of two composition operators on the Bergman spaces over the unit disc is characterized in [11] in terms of certain cancellation property of the inducing maps at every "bad" boundary points, which make each single composition operator not to be compact. In this paper, we completely characterize the compactness of a linear combination of three composition operators on the Bergman space. As one consequence of this characterization, we show that there is no cancellation property for the compactness of double difference of composition operators. More precisely, we show that if ϕi are distinct and none of Cϕi is compact, then (Cϕ1-Cϕ2)-(Cϕ3-Cϕ1) is compact if and only if both (Cϕ1-Cϕ2) and (Cϕ3-Cϕ1) are compact.

KW - Compactness

KW - Difference of composition operators

KW - Linear combination

UR - http://www.scopus.com/inward/record.url?scp=84939257434&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84939257434&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2015.07.027

DO - 10.1016/j.jmaa.2015.07.027

M3 - Article

AN - SCOPUS:84939257434

VL - 432

SP - 1174

EP - 1182

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -