Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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ON THE CLOSED RANGE COMPOSITION AND WEIGHTED COMPOSITION OPERATORS

  • Received : 2018.11.16
  • Accepted : 2019.03.13
  • Published : 2020.01.31
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Abstract

Let ψ be an analytic function on 𝔻, the unit disc in the complex plane, and φ be an analytic self-map of 𝔻. Let 𝓑 be a Banach space of functions analytic on 𝔻. The weighted composition operator Wφ,ψ on 𝓑 is defined as Wφ,ψf = ψf ◦ φ, and the composition operator Cφ defined by Cφf = f ◦ φ for f ∈ 𝓑. Consider α > -1 and 1 ≤ p < ∞. In this paper, we prove that if φ ∈ H∞(𝔻), then Cφ has closed range on any weighted Dirichlet space 𝒟α if and only if φ(𝔻) satisfies the reverse Carleson condition. Also, we investigate the closed rangeness of weighted composition operators on the weighted Bergman space Apα.

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References

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