Weighted composition operators on Korenblum type spaces of analytic functions
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Weighted composition operators on Korenblum type spaces of analytic functions Esther Gómez-Orts1 Received: 4 February 2020 / Accepted: 13 August 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract We investigate the continuity, compactness and invertibility of weighted composition operators Wψ,ϕ : f → ψ( f ◦ ϕ) when they act on the classical Korenblum space A−∞ and other related Fréchet or (LB)-spaces of analytic functions on the open unit disc which are defined as intersections or unions of weighted Banach spaces with sup-norms. Some results about the spectrum of these operators are presented in case the self-map ϕ has a fixed point in the unit disc. A precise description of the spectrum is obtained in this case when the operator acts on the Korenblum space. Keywords Weighted composition operator · Compact operator · Spectrum · analytic functions · Growth Banach spaces · Korenblum space · Fréchet spaces · (LB)-spaces Mathematics Subject Classification Primary 47B33 · Secondary 46A04 · 46E15 · 47B07 · 47B38
1 Introduction Let ϕ : D → D be an analytic self-map on the unit disc D of the complex plane, and let ψ : D → C be an analytic map. The aim of this article is to investigate the continuity, compactness, invertibility and the spectrum of weighted composition operators Wψ,ϕ : f → ψ( f ◦ϕ) when they act on certain weighted Fréchet or (LB)-spaces of analytic functions on D of the complex plane. The operator Wψ,ϕ can be written as the composition Wψ,ϕ = Mψ ◦Cϕ , where Mψ f := ψ f is the multiplication operator and Cϕ f := f ◦ ϕ is the composition operator. In this paper the operator Wψ,ϕ acts on spaces which appear as intersections or unions of the growth Banach spaces of analytic functions defined below. Weighted composition operators have been investigated by many authors. We refer to the books by Cowen and
This research was partially supported by the research project MTM2016-76647-P and the grant BES-2017-081200.
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Esther Gómez-Orts [email protected] Instituto Universitario de Matemática Pura y Aplicada IUMPA, Universitat Politècnica de València, 46022 València, Spain 0123456789().: V,-vol
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MacCluer [14] and Shapiro [26]. We also refer the reader to the articles [6,7,10,12,15,16,24] and the references therein. The space H (D) of all analytic functions on D is endowed with the Fréchet topology of uniform convergence on compact sets. When we write a space, we mean a Hausdorff locally convex space. We refer the reader to [23] for results and terminology about functional analysis, and in particular about Fréchet and (LB)-spaces. For each α > 0, the growth Banach spaces of analytic functions are defined as Hα∞ :=
f ∈ H (D) : f α := sup(1 − |z|)α | f (z)| < ∞ z∈D
and
Hα0 :=
f ∈ H (D) :
α
lim (1 − |z|) | f (z)| = 0 .
|z|→1−
These spaces are sometimes defined using the weight (1 − |z|2 )α instead of (1 − |z|)α . Since 1 − |z| ≤ 1 − |z|2 ≤ 2(1 − |z|), the spaces coincide and the norms are equivalent. Both Hα∞ and Hα0 are Banach spaces when endowed
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