Illinois Journal of Mathematics

Operator-weighted composition operators on vector-valued analytic function spaces

Jussi Laitila and Hans-Olav Tylli

Full-text: Open access

Abstract

We study qualitative properties of the operator-\break weighted composition maps ${W_{\psi,\varphi}} : f\mapsto\psi(f\circ\varphi)$ on the vector-valued spaces $H^\infty_v(X)$ of $X$-valued analytic functions $f : {\mathbb{D}}\to X$, where ${\mathbb{D}}$ is the unit disk, $X$ is a complex Banach space, $\varphi$ is an analytic self-map of ${\mathbb{D}}$, $\psi$ is an analytic operator-valued function on ${\mathbb{D}}$, and $v$ is a bounded continuous weight on ${\mathbb{D}}$. Boundedness and compactness properties of ${W_{\psi,\varphi}}$ are characterized on $H^\infty_v(X)$ for infinite-dimensional $X$. It turns out that the (weak) compactness of ${W_{\psi,\varphi}}$ also involves properties of the auxiliary operator $T_\psi : x \mapsto\psi(\cdot)x$ from $X$ to $H^\infty_v(X)$, in contrast to the familiar scalar-valued setting $X = \mathbb C$.

Article information

Source
Illinois J. Math., Volume 53, Number 4 (2009), 1019-1032.

Dates
First available in Project Euclid: 22 November 2010

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1290435336

Digital Object Identifier
doi:10.1215/ijm/1290435336

Mathematical Reviews number (MathSciNet)
MR2741175

Zentralblatt MATH identifier
1207.47021

Subjects
Primary: 47B33: Composition operators
Secondary: 46E40: Spaces of vector- and operator-valued functions

Citation

Laitila, Jussi; Tylli, Hans-Olav. Operator-weighted composition operators on vector-valued analytic function spaces. Illinois J. Math. 53 (2009), no. 4, 1019--1032. doi:10.1215/ijm/1290435336. https://projecteuclid.org/euclid.ijm/1290435336


Export citation

References

  • K. D. Bierstedt, J. Bonet and J. Taskinen, Associated weights and spaces of holomorphic functions, Studia Math. 127 (1998), 137–168.
  • J. Bonet, P. Domański and M. Lindström, Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions, Canad. Math. Bull. 42 (1999), 139–148.
  • J. Bonet, P. Domański and M. Lindström, Pointwise multiplication operators on weighted Banach spaces of analytic functions, Studia Math. 137 (1999), 177–194.
  • J. Bonet, P. Domański and M. Lindström, Weakly compact composition operators on analytic vector-valued function spaces, Ann. Acad. Sci. Fenn. Math. 26 (2001), 233–248.
  • J. Bonet, P. Domański, M. Lindström and J. Taskinen, Composition operators between weighted Banach spaces of analytic functions, J. Austral. Math. Soc. Ser. A 64 (1998), 101–118.
  • J. Bonet and M. Friz, Weakly compact composition operators on locally convex spaces, Math. Nachr. 245 (2002), 26–44.
  • M. Cambern and K. Jarosz, Multipliers and isometries in $H\sp\infty\sb E$, Bull. London Math. Soc. 22 (1990), 463–466.
  • D. M. Campbell and R. J. Leach, A survey of $H^p$ multipliers as related to classical function theory, Complex Var. 3 (1984), 85–111.
  • M. D. Contreras and S. Díaz-Madrigal, Compact-type operators defined on $H^\infty$, Function spaces (Edwardsville, IL, 1998), Contemp. Math., vol. 232, Amer. Math. Soc., Providence, 1999, pp. 111–118.
  • M. D. Contreras and A. G. Hernández-Díaz, Weighted composition operators in weighted Banach spaces of analytic functions, J. Austral. Math. Soc. 69 (2000), 41–60.
  • M. D. Contreras and A. G. Hernández-Díaz, Weighted composition operators on Hardy spaces, J. Math. Anal. Appl. 263 (2001), 224–233.
  • C. C. Cowen and B. D. MacCluer, Composition operators on spaces of analytic functions, CRC Press, Boca Raton, FL, 1995.
  • \uZ. \uCučković and R. Zhao, Weighted composition operators on the Bergman space, J. London Math. Soc. 70 (2004), 499–511.
  • J. Laitila, Weakly compact composition operators on vector-valued $\mathit{BMOA}$, J. Math. Anal. Appl. 308 (2005), 730–745.
  • J. Laitila, Composition operators and vector-valued $\mathit{BMOA}$, Integral Equations Operator Theory 58 (2007), 487–502.
  • J. Laitila and H.-O. Tylli, Composition operators on vector-valued harmonic functions and Cauchy transforms, Indiana Univ. Math. J. 55 (2006), 719–746.
  • J. Laitila, H.-O. Tylli and M. Wang, Composition operators from weak to strong spaces of vector-valued analytic functions, J. Operator Theory 62 (2009), 281–295.
  • P.-K. Lin, The isometries of $H^\infty(E)$, Pacific J. Math. 143 (1990), 69–77.
  • P. Liu, E. Saksman and H.-O. Tylli, Small composition operators on analytic vector-valued function spaces, Pacific J. Math. 184 (1998), 295–309.
  • J. S. Manhas, Multiplication operators on weighted locally convex spaces of vector-valued analytic functions, Southeast Asian Bull. Math. 27 (2003), 649–660.
  • A. Montes-Rodríguez, Weighted composition operators on weighted Banach spaces of analytic functions, J. London Math. Soc. 61 (2000), 872–884.
  • A. Pietsch, Operator ideals, North-Holland, Amsterdam, 1980.
  • J. H. Shapiro, Composition operators and classical function theory, Springer-Verlag, New York, 1993.
  • J. Taskinen, Compact composition operators on general weighted spaces, Houston J. Math. 27 (2001), 203–218.
  • P. Wojtaszczyk, Banach spaces for analysts, Cambridge Univ. Press, Cambridge, 1991.
  • K. Zhu, Operator theory in function spaces, Marcel Dekker, New York, 1990.