Commutators of weighted composition operators

Sungeun Jung, Yoenha Kim, Eungil Ko

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1 Scopus citations


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


  • Commutant
  • Commutator
  • Compact operator
  • Weighted composition operator


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