Taiwanese Journal of Mathematics


Zhi Jie Jiang

Full-text: Open access


Let $\mathbb D = \{ z \in \mathbb C: |z| \lt 1\}$ be the open unit disk in the complex plane $\mathbb C$, $H(\mathbb D)$ be the space of all analytic functions on $\mathbb D$, $\varphi$ be an analytic self-map of $\mathbb D$ and $u \in H(\mathbb D)$. Define operators by $DW_{\varphi,u}f = (u \cdot f \circ \varphi)'$ and $W_{\varphi,u}Df = (u \cdot f' \circ \varphi)$ for $f \in H(\mathbb D)$. In this paper we characterize bounded operators $DW_{\varphi,u}$ and $W_{\varphi,u}D$ from weighted Bergman space to Zygmund-type space, Bloch-type space and Bers-type space on the open unit disk. We also give some sufficient and necessary conditions for these operators to be compact operators in terms of inducing maps $\varphi$ and $u$.

Article information

Taiwanese J. Math., Volume 15, Number 5 (2011), 2095-2121.

First available in Project Euclid: 18 July 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47B38: Operators on function spaces (general)
Secondary: 47B33: Composition operators 47B37: Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)

weighted Bergman space Bers-type space Bloch-type space Zygmund-type space weighted composition operator


Jiang, Zhi Jie. ON A CLASS OF OPERATORS FROM WEIGHTED BERGMAN SPACES TO SOME SPACES OF ANALYTIC FUNCTIONS. Taiwanese J. Math. 15 (2011), no. 5, 2095--2121. doi:10.11650/twjm/1500406425. https://projecteuclid.org/euclid.twjm/1500406425

Export citation


  • K. D. Bierstedt and W. H. Summers, Biduals of weighted Banach spaces of analytic functions, J. Austral. Math. Soc. $($Series A$)$, 54 (1993), 70-79.
  • J. S. Choa and S. Ohno, Product of composition and analytic Toeplitz operators, J. Math. Anal. Appl., \bf281 (2003), 320-332.
  • C. C. Cowen and B. D. MacCluer, Composition operators on spaces of analytic functions, CRC Press, 1995.
  • Z. J. Jiang and H. B. Bai, Weighted composition operator on Hardy space $H^{p}(B_{N})$ (in Chinese), Advances In Mathematics, \bf37(6) (2008), 749-754.
  • W. X. He and Y. Z. Li, Bers-type spaces and composition operators, Acta Northeast Math J, \bf18(3) (2002), 223-232.
  • W. X. He and L. J. Jiang, Composition operator on Bers-type spaces, Acta Mathematica Sientia, \bf22B(3) (2002), 404-412.
  • R. A. Hibschweiler and N. Portnoy, Composition followed by differentiation between Bergman and Hardy spaces, Rocky Mountain J. Math., \bf35(3) (2005), 843-855.
  • S. Li and S. Stević, Volterra-type operators on Zygmund spaces, Journal of Inequalities and Applications, Vol. 2007, Article ID 32124, 2007, 10 pages.
  • S. Li and S. Stević, Weighted composition operators from Zygmund spaces into Bloch spaces, Appl. Math. Comput., 206(2) (2008), 825-831.
  • S. Li and S. Stević, Integral-type operators from Bloch-type spaces to Zygmund-type spaces, Appl. Math. Comput., 215 (2009), 464-473.
  • S. Ohno, Weighted composition operators between $H^{\infty}$ and the Bloch space, Taiwan. J. Math. Soc., 5(3) (2001), 555-563.
  • S. Ohno, Products of composition and differentiation on Bloch spaces, Bull. Korean Math. Soc., 46(6) (2009), 1135-1140.
  • S. Stević, Norm of weighted composition operators from Bloch space to $H_{\mu}^{\infty}$ on the unit ball, Ars. Combin., 88 (2008), 125-127.
  • S. Stevi\' c, On an integral operator from the Zygmund space to the Bloch-type space on the unit ball, Glasg. J. Math., 51 (2009), 275-287.
  • W. Yang, Weighted composition operators from Bloch-type spaces to weighted-type spaces, Ars. Combin., 93 (2009), 265-274.
  • K. Zhu, Spaces of holomorphic functions in the unit ball, Springer, New York, 2005.
  • K. Zhu, Operator theory in function space, Dekker, New York, 1990.
  • X. Zhu, Weighted composition operators from logarithmic Bloch spaces to a class of weighted-type spaces in the unit ball, Ars. Combin., 91 (2009), 87-95.
  • X. Zhu, Weighted composition operators from $F(p,q,s)$ spaces to $H^\infty_\mu$ spaces, Abstr. Appl. Anal., Vol. 2009, Article ID 290978, (2009), 12 pages.