TY - JOUR
T1 - Self-commutators of invertible weighted composition operators on H2
AU - Jung, Sungeun
AU - Ko, Eungil
N1 - Funding Information:
The authors wish to thank the referee for a careful reading and valuable comments for the original draft. This work was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) grant funded by the Korea government(MEST)(2012-0000939).
PY - 2014/5
Y1 - 2014/5
N2 - In this paper, we consider the self-commutator of an invertible weighted composition operator Wf, φ on the Hardy space H2 where f is continuous on D̄. We show that both the self-commutator [Wf,φ*, Wf,φ] and the anti-self-commutator {Wf,φ*, Wf,φ} are expressed as compact perturbations of Toeplitz operators. Moreover, we give an alternative proof for the result in [2] that Wf,φ is unitary exactly when φ is an automorphism of D and (Formula presented.) where p = φ-1(0), Kp is the reproducing kernel at p for H2, and c is a constant with {pipe}c{pipe} = 1. We next show that when {pipe}f(z){pipe} ≤ {pipe}f(φ(z)){pipe} for all z ∈ D, the weighted composition operator Wf,φ is normal if and only if the composition operator Cφ is unitary and f is constant on D. We also provide some spectral properties of Wf,φ* Wf,φ and Wf,φ Wf,φ*.
AB - In this paper, we consider the self-commutator of an invertible weighted composition operator Wf, φ on the Hardy space H2 where f is continuous on D̄. We show that both the self-commutator [Wf,φ*, Wf,φ] and the anti-self-commutator {Wf,φ*, Wf,φ} are expressed as compact perturbations of Toeplitz operators. Moreover, we give an alternative proof for the result in [2] that Wf,φ is unitary exactly when φ is an automorphism of D and (Formula presented.) where p = φ-1(0), Kp is the reproducing kernel at p for H2, and c is a constant with {pipe}c{pipe} = 1. We next show that when {pipe}f(z){pipe} ≤ {pipe}f(φ(z)){pipe} for all z ∈ D, the weighted composition operator Wf,φ is normal if and only if the composition operator Cφ is unitary and f is constant on D. We also provide some spectral properties of Wf,φ* Wf,φ and Wf,φ Wf,φ*.
KW - self-commutator
KW - Toeplitz operator
KW - weighted composition operator
UR - http://www.scopus.com/inward/record.url?scp=84897100940&partnerID=8YFLogxK
U2 - 10.1080/17476933.2013.778837
DO - 10.1080/17476933.2013.778837
M3 - Article
AN - SCOPUS:84897100940
SN - 1747-6933
VL - 59
SP - 693
EP - 705
JO - Complex Variables and Elliptic Equations
JF - Complex Variables and Elliptic Equations
IS - 5
ER -