Abstract
In the setting of the Hardy spaces or the standard weighted Bergman spaces over the unit ball in Cn, linear fractional composition operators are known to behave quite rigidly in the sense that they cannot form any nontrivial compact differences or, more generally, linear combinations. In this paper, in the setting of the standard weighted Bergman spaces over the half-plane, we completely characterize bounded/compact differences of linear fractional composition operators. Our characterization reveals that a linear fractional composition operator can possibly form a compact difference, which is a new half-plane phenomenon due to the half-plane not being bounded. Also, we obtain necessary conditions and sufficient conditions for a linear combination to be bounded/compact. As a consequence, when the weights and exponents of the weighted Bergman spaces are restricted to a certain range, we obtain a characterization for a linear combination to be bounded/compact. Applying our results, we provide an example showing a double difference cancellation phenomenon for linear combinations of three linear fractional composition operators, which is yet another half-plane phenomenon.
| Original language | English |
|---|---|
| Article number | 27 |
| Journal | Integral Equations and Operator Theory |
| Volume | 90 |
| Issue number | 3 |
| DOIs | |
| Publication status | Published - 2018 Jun 1 |
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Keywords
- Bounded operator
- Compact operator
- Difference
- Half-plane
- Hilbert–Schmidt operator
- Linear combination
- Linear fractional composition operator
- Weighted Bergman space
ASJC Scopus subject areas
- Analysis
- Algebra and Number Theory
Cite this
Linear Fractional Composition Operators Over the Half-Plane. / Choe, Boo Rim; Koo, Hyung Woon; Smith, Wayne.
In: Integral Equations and Operator Theory, Vol. 90, No. 3, 27, 01.06.2018.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Linear Fractional Composition Operators Over the Half-Plane
AU - Choe, Boo Rim
AU - Koo, Hyung Woon
AU - Smith, Wayne
PY - 2018/6/1
Y1 - 2018/6/1
N2 - In the setting of the Hardy spaces or the standard weighted Bergman spaces over the unit ball in Cn, linear fractional composition operators are known to behave quite rigidly in the sense that they cannot form any nontrivial compact differences or, more generally, linear combinations. In this paper, in the setting of the standard weighted Bergman spaces over the half-plane, we completely characterize bounded/compact differences of linear fractional composition operators. Our characterization reveals that a linear fractional composition operator can possibly form a compact difference, which is a new half-plane phenomenon due to the half-plane not being bounded. Also, we obtain necessary conditions and sufficient conditions for a linear combination to be bounded/compact. As a consequence, when the weights and exponents of the weighted Bergman spaces are restricted to a certain range, we obtain a characterization for a linear combination to be bounded/compact. Applying our results, we provide an example showing a double difference cancellation phenomenon for linear combinations of three linear fractional composition operators, which is yet another half-plane phenomenon.
AB - In the setting of the Hardy spaces or the standard weighted Bergman spaces over the unit ball in Cn, linear fractional composition operators are known to behave quite rigidly in the sense that they cannot form any nontrivial compact differences or, more generally, linear combinations. In this paper, in the setting of the standard weighted Bergman spaces over the half-plane, we completely characterize bounded/compact differences of linear fractional composition operators. Our characterization reveals that a linear fractional composition operator can possibly form a compact difference, which is a new half-plane phenomenon due to the half-plane not being bounded. Also, we obtain necessary conditions and sufficient conditions for a linear combination to be bounded/compact. As a consequence, when the weights and exponents of the weighted Bergman spaces are restricted to a certain range, we obtain a characterization for a linear combination to be bounded/compact. Applying our results, we provide an example showing a double difference cancellation phenomenon for linear combinations of three linear fractional composition operators, which is yet another half-plane phenomenon.
KW - Bounded operator
KW - Compact operator
KW - Difference
KW - Half-plane
KW - Hilbert–Schmidt operator
KW - Linear combination
KW - Linear fractional composition operator
KW - Weighted Bergman space
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UR - http://www.scopus.com/inward/citedby.url?scp=85046258734&partnerID=8YFLogxK
U2 - 10.1007/s00020-018-2450-x
DO - 10.1007/s00020-018-2450-x
M3 - Article
AN - SCOPUS:85046258734
VL - 90
JO - Integral Equations and Operator Theory
JF - Integral Equations and Operator Theory
SN - 0378-620X
IS - 3
M1 - 27
ER -