Self-commutators of invertible weighted composition operators on H2

Sungeun Jung, Eungil Ko

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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*.

Original languageEnglish
Pages (from-to)693-705
Number of pages13
JournalComplex Variables and Elliptic Equations
Issue number5
StatePublished - May 2014


  • self-commutator
  • Toeplitz operator
  • weighted composition operator


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