Banach Journal of Mathematical Analysis

Volterra composition operators on logarithmic Bloch spaces

Xiangling Zhu

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Abstract

Let $\varphi$ be a holomorphic self-map and $g$ a fixed holomorphic function on the unit ball $B$. The boundedness and compactness of the Volterra composition operator $$T_{g,\varphi} f(z)= \int_0^1 f(\varphi(tz)) \Re g(tz)\frac{dt}{t}$$ on the logarithmic Bloch space and little logarithmic Bloch space are studied in this paper.

Article information

Source
Banach J. Math. Anal., Volume 3, Number 1 (2009), 122-130.

Dates
First available in Project Euclid: 21 April 2009

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1240336429

Digital Object Identifier
doi:10.15352/bjma/1240336429

Mathematical Reviews number (MathSciNet)
MR2461752

Zentralblatt MATH identifier
1163.47022

Subjects
Primary: 47B38: Operators on function spaces (general)
Secondary: 32A37: Other spaces of holomorphic functions (e.g. bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) [See also 46Exx]

Keywords
Volterra composition operator logarithmic Bloch space Bloch space

Citation

Zhu, Xiangling. Volterra composition operators on logarithmic Bloch spaces. Banach J. Math. Anal. 3 (2009), no. 1, 122--130. doi:10.15352/bjma/1240336429. https://projecteuclid.org/euclid.bjma/1240336429


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References

  • D. Chang, S. Li and S. Stevi\' c, On some integral operators on the unit polydisk and the unit ball, Taiwanese J. Math. 11 (5) (2007), 1251–1286.
  • C.C. Cowen and B.D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995.
  • Z. Hu, Extended Cesàro operators on mixed norm spaces, Proc. Amer. Math. Soc. 131 (7) (2003), 2171–2179.
  • Z. Hu, Extended Cesàro operators on Bergman spaces, J. Math. Anal. Appl. 296 (2004), 435–454.
  • S. Li, Volterra composition operators between weighted Bergman space and Bloch type spaces, J. Korea Math. Soc. 45(1) (2008), 229–248.
  • S. Li, Riemann-Stieltjes operators from $F(p,q,s)$ to Bloch space on the unit ball, J. Inequal. Appl. 2006, Art. ID 27874, 14 pp.
  • S. Li and S. Stevi\' c, Riemann-Stieltjes type integral operators on the unit ball in $\mathbbC^n$, Complex Var. Elliptic Equ. 52 (6) (2007), 495–517.
  • S. Li and S. Stevi\' c, Riemann-Stieltjes operators on Hardy spaces in the unit ball of $\CC^n$, Bull. Belg. Math. Soc. Simon Stevin, 14 (2007), 621–628.
  • S. Li and S. Stevi\' c, Riemann-Stieltjes operators between different weighted Bergman spaces, Bull. Belg. Math. Soc. Simon Stevin, 15 (2008), 677–686.
  • K. Madigan and A. Matheson, Compact composition operators on the Bloch space, Trans. Amer. Math. Soc. 347 (7) (1995), 2679–2687.
  • S. Stević, On an integral operator on the unit ball in $\Bbb C\sp n$, J. Inequal. Appl. 1 (2005), 81–88.
  • J. Xiao, Riemann-Stieltjes operators on weighted Bloch and Bergman spaces of the unit ball, J. London. Math. Soc. 70 (2) (2004), 199–214.
  • K. Zhu, Multipliers of BMO in the Bergman metric with applications to Toeplitz operators, J. Funct. Anal. 87 (1989), 31–50.
  • K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Springer-Verlag, New York, 2005.
  • X. Zhu, Volterra composition operators from generalized weighted Bergman spaces to $\mu$-Bloch type spaces, J. Funct. Space Appl., to appear.