Commutators of weighted composition operators

Sungeun Jung, Yoenha Kim, Eungil Ko

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In this paper, we prove that if the composition symbols φ and ψ are linear fractional non-automorphisms of such that φ(ζ) and ψ(ζ) belong to ∂ for some ζ ∈ ∂ and u, v ∈ H are continuous on ∂ with u(ζ)v(ζ) ≠ 0, then [{W*v, psi, Wu,φ] is compact on H2 if and only if ζ is the common boundary fixed point of φ and ψ and one of the following statements holds: (i) both φ and ψ are parabolic; (ii) both φ and ψ are hyperbolic and another fixed point of φ is 1=w where w is the fixed point of ψ other than ζ. We also study the commutant of a weighted composition operator on H2. We verify that if φ is an analytic self-map of with Denjoy-Wolff point b ∈ and u ∈ H\{0}, then every weighted composition operator in the commutant {Wu, φ}′ has {f ∈ H2 : f(b) = 0} as its nontrivial invariant subspace.

Original languageEnglish
Article number1450053
JournalInternational Journal of Mathematics
Issue number6
StatePublished - Jun 2014

Bibliographical note

Funding Information:
The authors wish to thank the referee for a careful reading and valuable comments for the original draft. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093827).


  • Commutant
  • Commutator
  • Compact operator
  • Weighted composition operator


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