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Let $\psi$ be an analytic function on $\D$, the unit disc in the complex plane, and $\varphi$ be an analytic self-map of $\D $. Let $\mathcal{B}$ be a Banach space of functions analytic on $\D$. The weighted composition operator $\W $ on $\mathcal{B}$ is defined as $\W f=\psi f\circ \varphi$, and the composition operator $\C$ defined by $\C f=f\circ \varphi$ for $f\in \mathcal{B}$. Consider $\alpha >-1$ and $1\leq p<\infty$. In this paper, we prove that if $\varphi\in H^\infty(\D)$, then $\C$ has closed range on any weighted Dirichlet space $\d$ if and only if $\varphi(\D)$ satisfies the reverse Carleson condition. Also, we investigate the closed rangeness of weighted composition operators on the weighted Bergman space $\A$.
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참고문헌 (13)
참고문헌 신청
J. R. Akeroyd / 2017 / A note on closed-range composition operators / Complex Anal. Oper. Theory 11(1):151~160
J. R. Akeroyd / 2012 / Closed-range composition operators on weighted Bergman spaces / Integral Equations Operator Theory 72(1):103~114
J. R. Akeroyd / 2008 / Closed-range composition operators on A2 / Illinois J. Math 52(2):533~549
J. A. Cima / 1974 / On some properties of composition operators / Indiana Univ. Math. J. 24:215~220
C. C. Cowen / 1995 / Composition operators on spaces of analytic functions, Studies in Advanced Mathematics / CRC Press
J. R. Akeroyd / 2012 / Closed-range composition operators on weighted Bergman spaces / Integral Equations Operator Theory 72(1):103~114
J. R. Akeroyd / 2008 / Closed-range composition operators on A2 / Illinois J. Math 52(2):533~549
J. A. Cima / 1974 / On some properties of composition operators / Indiana Univ. Math. J. 24:215~220
C. C. Cowen / 1995 / Composition operators on spaces of analytic functions, Studies in Advanced Mathematics / CRC Press
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