### Abstract

In the setting of the Hardy spaces or the standard weighted Bergman spaces over the unit ball in C^{n}, 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.

Research output: Contribution to journal › Article

*Integral Equations and Operator Theory*, vol. 90, no. 3, 27. https://doi.org/10.1007/s00020-018-2450-x

}

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

UR - http://www.scopus.com/inward/record.url?scp=85046258734&partnerID=8YFLogxK

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

VL - 90

JO - Integral Equations and Operator Theory

JF - Integral Equations and Operator Theory

SN - 0378-620X

IS - 3

M1 - 27

ER -