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http://dx.doi.org/10.4134/CKMS.c180474

ON THE CLOSED RANGE COMPOSITION AND WEIGHTED COMPOSITION OPERATORS  

Keshavarzi, Hamzeh (Department of Mathematics College of Sciences Shiraz University)
Khani-Robati, Bahram (Department of Mathematics College of Sciences Shiraz University)
Publication Information
Communications of the Korean Mathematical Society / v.35, no.1, 2020 , pp. 217-227 More about this Journal
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.
Keywords
Composition operators; weighted Dirichlet spaces; weighted composition operators; weighted Bergman spaces; closed range; reverse Carleson condition;
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