Rocky Mountain Journal of Mathematics

Composition operators on weighted Hardy spaces

Waleed Al-Rawashdeh

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Suppose $\varphi$ is an analytic self-map of open unit disk $\mathbf{D}$ and $\psi$ is an analytic function on $\mathbf{D}$. Then a weighted composition operator induced by $\varphi$ with weight $\psi$ is given by $(W_{\psi, \varphi}f)(z)= \psi(z)f(\varphi(z))$, for $z \in \mathbf{D}$ and $f$ analytic on $\mathbf{D}$. Necessary and sufficient conditions are given for the boundedness and compactness of the weighted composition operators $W_{\psi, \varphi}$. In terms of fixed points in the closed unit disk $\overline{\mathbf{D}}$, conditions under which $W_{\psi, \varphi}$ is compact are given. Necessary conditions for the compactness of $C_{\varphi}$ are given in terms of the angular derivative $\varphi^{\prime}(\zeta)$ where $\zeta$ is on the boundary of the unit disk. Moreover, we present sufficient conditions for the membership of composition operators in the Schatten $p$-class $S_{p}(H^s(\beta_1), H^q(\beta_2))$, where the inducing map has supremum norm strictly smaller than~$1$.

Article information

Rocky Mountain J. Math., Volume 44, Number 4 (2014), 1053-1072.

First available in Project Euclid: 31 October 2014

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Zentralblatt MATH identifier

Primary: 47B38: Operators on function spaces (general)
Secondary: 47B15: Hermitian and normal operators (spectral measures, functional calculus, etc.) 47B33: Composition operators

Weighted composition operators compact operator angular derivative Schatten $p$-class weighted Hardy spaces


Al-Rawashdeh, Waleed. Composition operators on weighted Hardy spaces. Rocky Mountain J. Math. 44 (2014), no. 4, 1053--1072. doi:10.1216/RMJ-2014-44-4-1053.

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  • C.C. Cowen and B.D. MacCluer, Composition operators on spaces of analytic functions, CRC press, Boca Raton, 1995.
  • B.D. MacCluer, X. Zeng and N. Zorboska, Composition operators on small weighted Hardy spaces, Illinois J. Math. 40 (1996), 662–667.
  • J.R. Ringrose, Compact non-self-adjoint operators, Van Nostrand-Reinhold, New York, 1971.
  • R. Schatten, Norm ideals of completely continuous operators, Springer-Verlag, Berlin, 1960.
  • J.H. Shapiro, Composition operators on spaces of boundary-regular holomorphic functions, Proc. Amer. Math. Soc. 100 (1987), 49–57.
  • B. Yousefi, Composition operators on weighted Hardy spaces, Kyungpook Math. J. 44 (2004), 319–324.
  • B. Yousefi and M. Ahmadian, Hypercylic and compact composition operators on Banach spaces of formal power series, Inter. Math. Forum 27 (2008), 1347–1353.
  • N. Zorboska, Composition operators induced by functions with supremum strictly smaller than 1, Proc. Amer. Math. Soc. 106 (1989), 679–684.
  • N. Zorboska, Compact composition operators on some weighted Hardy spaces, J. Oper. Theor. 22 (1989), 233–241.