## 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 |

### Bibliographical note

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).

## Keywords

- Toeplitz operator
- self-commutator
- weighted composition operator

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