TY - JOUR

T1 - Linear Fractional Composition Operators Over the Half-Plane

AU - Choe, Boo Rim

AU - Koo, Hyungwoon

AU - Smith, Wayne

N1 - Funding Information:
We thank the referee for his/her drawing our attention to the reference [24]. B. R. Choe was supported by NRF (2015R1D1A1A01057685) of Korea and H. Koo was supported by NRF (2017R1A2B2002515) of Korea.
Publisher Copyright:
© 2018, Springer International Publishing AG, part of Springer Nature.

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

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 -