Missouri Journal of Mathematical Sciences

Composition Operators on Generalized Weighted Nevanlinna Class

Waleed Al-Rawashdeh

Full-text: Open access

Abstract

Let $\varphi$ be an analytic self-map of open unit disk $\mathbb{D}$. The operator given by $(C_{\varphi}f)(z)=f(\varphi(z))$, for $z \in \mathbb{D}$ and $f$ analytic on $\mathbb{D}$ is called a composition operator. Let $\omega$ be a weight function such that $\omega\in L^{1}(\mathbb{D}, dA)$. The space we consider is a generalized weighted Nevanlinna class $\mathcal{N}_{\omega}$, which consists of all analytic functions $f$ on $\mathbb{D}$ such that $\displaystyle \|f\|_{\omega}=\int_{\mathbb{D}}\log^{+}(|f(z)|) \omega(z) dA(z)$ is finite; that is, $\mathcal{N}_{\omega}$ is the space of all analytic functions belong to $L_{\log^+}(\mathbb{D}, \omega dA)$. In this paper we investigate, in terms of function-theoretic, composition operators on the space $\mathcal{N}_{\omega}$. We give sufficient conditions for the boundedness and compactness of these composition operators.

Article information

Source
Missouri J. Math. Sci., Volume 26, Issue 1 (2014), 14-22.

Dates
First available in Project Euclid: 10 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.mjms/1404997105

Digital Object Identifier
doi:10.35834/mjms/1404997105

Mathematical Reviews number (MathSciNet)
MR3263539

Zentralblatt MATH identifier
1298.47046

Subjects
Primary: 47B38: Operators on function spaces (general)
Secondary: 47B15: Hermitian and normal operators (spectral measures, functional calculus, etc.) 47B33: Composition operators 30H05: Bounded analytic functions

Keywords
Composition operators generalized weighted Nevanlinna class weighted Bergman spaces compact operators bounded operators radial weight function

Citation

Al-Rawashdeh, Waleed. Composition Operators on Generalized Weighted Nevanlinna Class. Missouri J. Math. Sci. 26 (2014), no. 1, 14--22. doi:10.35834/mjms/1404997105. https://projecteuclid.org/euclid.mjms/1404997105


Export citation

References

  • J. C. Choa and H. O. Kim, Compact composition operators on the Nevanlinna class, Pro. Amer. Math. Soc., 125 (1997), 145–151.
  • J. C. Choa and H. O. Kim, On function-theoretic conditions characterizing compact composition operators on $H^2$, Pro. Japan Acad., 97 (1999), 109–112.
  • J. C. Choa, H. O. Kim, and J. H. Shapiro, Compact composition operators on the Smirnov class, Pro. Amer. Math. Soc., 128 (2000), 2297–2308.
  • B. R. Choe, H. Koo, and W. Smith, Carleson measure for the area Nevanlinna spaces and applications, J. D'Analyse Math., 104 (2008), 207–233.
  • C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC press, Boca Raton, 1995.
  • P. Duren and A. Schuster, Bergman Spaces, American Mathematical Society, 2004.
  • A. Haldimann and H. Jarchow, Nevanlinna algebras, Studia Math., 147 (2001), 243–268.
  • H. Hedenmalm, B. Korenblum, and K. Zhu, Theory of Bergman Spaces, Springer-Verlag, New York, 2000.
  • N. Jaoua, Order bounded composition operators on the Hardy spaces and the Nevanlinna class, Studia Math., 134 (1999), 35–55.
  • H. Jarchow, Locally Convex Spaces, Teubner-Verlag, Stuttgart, 1981.
  • H. Jarchow and J. Xiao, Composition operators between Nevanlinna classes and Bergman spaces with weights, J. Operator Theory, 46 (2001), 605–618.
  • N. J. Kalton, N. T. Peck and J. W. Roberts, An F-Spaces Sampler, London Math. Soc. Lecture Note Ser., Vol. 89, Cambridge Univ. Press, Cambridge, 1984.
  • T. L. Kriete and B. D. MacCluer, Composition operators on large weighted Bergman spaces, Indiana Univ. Math. J., 41 (1992), 755–788.
  • F. Morgan, Geometric Measure Theory. Academic Press, Boston, 1988.
  • J. H. Shapiro and A. L. Shields, Unusual topological properties of the Nevanlinna class, Amer. J. Math, 97 (1976), 915–936.
  • J. H. Shapiro, Composition Operators and Classical Function Theory. Universitext: Tracts in Mathematics. Springer-Verlag, New York, 1993.
  • J. Xiao, Compact composition operators on the area-Nevanlinna class, Exposition. Math., 17 (1999), 255–264.
  • K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Springer-Verlag, New York, 2005.
  • K. Zhu, Operator Theory in Function Spaces, Monographs and Textbooks in Pure and Applied Mathematics, 139, Marcel Dekker, Inc., New York, 1990.