## Abstract

In this paper, we consider the self-commutator of an invertible weighted composition operator W_{f, φ} on the Hardy space H^{2} where f is continuous on D̄. We show that both the self-commutator [W_{f,φ}*, W_{f,φ}] and the anti-self-commutator {W_{f,φ}*, W_{f,φ}} are expressed as compact perturbations of Toeplitz operators. Moreover, we give an alternative proof for the result in [2] that W_{f,φ} is unitary exactly when φ is an automorphism of D and (Formula presented.) where p = φ^{-1}(0), K_{p} is the reproducing kernel at p for H^{2}, 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 W_{f,φ} is normal if and only if the composition operator C_{φ} is unitary and f is constant on D. We also provide some spectral properties of W_{f,φ}* W_{f,φ} and W_{f,φ} W_{f,φ}*.

Original language | English |
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Pages (from-to) | 693-705 |

Number of pages | 13 |

Journal | Complex Variables and Elliptic Equations |

Volume | 59 |

Issue number | 5 |

DOIs | |

State | Published - May 2014 |

## Keywords

- self-commutator
- Toeplitz operator
- weighted composition operator

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