• Title, Summary, Keyword: generalized weighted composition operator

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GENERALIZED WEIGHTED COMPOSITION OPERATORS FROM AREA NEVANLINNA SPACES TO WEIGHTED-TYPE SPACES

  • Weifeng, Yang;Weiren, Yan
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.6
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    • pp.1195-1205
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    • 2011
  • Let $H(\mathbb{D})$ denote the class of all analytic functions on the open unit disk $\mathbb{D}$ of the complex plane $\mathbb{C}$. Let n be a nonnegative integer, ${\varphi}$ be an analytic self-map of $\mathbb{D}$ and $u{\in}H(\mathbb{D})$. The generalized weighted composition operator is defined by $$D_{{\varphi},u}^nf=uf^{(n)}{\circ}{\varphi},\;f{\in}H(\mathbb{D})$$. The boundedness and compactness of the generalized weighted composition operator from area Nevanlinna spaces to weighted-type spaces and little weighted-type spaces are characterized in this paper.

GENERALIZED COMPOSITION OPERATORS FROM GENERALIZED WEIGHTED BERGMAN SPACES TO BLOCH TYPE SPACES

  • Zhu, Xiangling
    • Journal of the Korean Mathematical Society
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    • v.46 no.6
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    • pp.1219-1232
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    • 2009
  • Let H(B) denote the space of all holomorphic functions on the unit ball B of $\mathbb{C}^n$. Let $\varphi$ = (${\varphi}_1,{\ldots}{\varphi}_n$) be a holomorphic self-map of B and $g{\in}2$(B) with g(0) = 0. In this paper we study the boundedness and compactness of the generalized composition operator $C_{\varphi}^gf(z)=\int_{0}^{1}{\mathfrak{R}}f(\varphi(tz))g(tz){\frac{dt}{t}}$ from generalized weighted Bergman spaces into Bloch type spaces.

HYPERCYCLICITY OF WEIGHTED COMPOSITION OPERATORS ON THE UNIT BALL OF ℂN

  • Chen, Ren-Yu;Zhou, Ze-Hua
    • Journal of the Korean Mathematical Society
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    • v.48 no.5
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    • pp.969-984
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    • 2011
  • This paper discusses the hypercyclicity of weighted composition operators acting on the space of holomorphic functions on the open unit ball $B_N$ of $\mathbb{C}^N$. Several analytic properties of linear fractional self-maps of $B_N$ are given. According to these properties, a few necessary conditions for a weighted composition operator to be hypercyclic in the space of holomorphic functions are proved. Besides, the hypercyclicity of adjoint of weighted composition operators are studied in this paper.

WEIGHTED COMPOSITION OPERATORS ON NACHBIN SPACES WITH OPERATOR-VALUED WEIGHTS

  • Klilou, Mohammed;Oubbi, Lahbib
    • Communications of the Korean Mathematical Society
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    • v.33 no.4
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    • pp.1125-1140
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    • 2018
  • Let A be a normed space, ${\mathcal{B}}(A)$ the algebra of all bounded operators on A, and V a family of strongly upper semicontinuous functions from a Hausdorff completely regular space X into ${\mathcal{B}}(A)$. In this paper, we investigate some properties of the weighted spaces CV (X, A) of all A-valued continuous functions f on X such that the mapping $x{\mapsto}v(x)(f(x))$ is bounded on X, for every $v{\in}V$, endowed with the topology generated by the seminorms ${\parallel}f{\parallel}v={\sup}\{{\parallel}v(x)(f(x)){\parallel},\;x{\in}X\}$. Our main purpose is to characterize continuous, bounded, and locally equicontinuous weighted composition operators between such spaces.