Abstract and Applied Analysis

On the Norm of Certain Weighted Composition Operators on the Hardy Space

M. Haji Shaabani and B. Khani Robati

Full-text: Open access

Abstract

We obtain a representation for the norm of certain compact weighted composition operator C ψ , φ on the Hardy space H 2 , whenever φ ( z ) = a z + b and ψ ( z ) = a z b . We also estimate the norm and essential norm of a class of noncompact weighted composition operators under certain conditions on φ and ψ . Moreover, we characterize the norm and essential norm of such operators in a special case.

Article information

Source
Abstr. Appl. Anal., Volume 2009 (2009), Article ID 720217, 13 pages.

Dates
First available in Project Euclid: 16 March 2010

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1268745584

Digital Object Identifier
doi:10.1155/2009/720217

Mathematical Reviews number (MathSciNet)
MR2533571

Zentralblatt MATH identifier
1176.47023

Citation

Haji Shaabani, M.; Khani Robati, B. On the Norm of Certain Weighted Composition Operators on the Hardy Space. Abstr. Appl. Anal. 2009 (2009), Article ID 720217, 13 pages. doi:10.1155/2009/720217. https://projecteuclid.org/euclid.aaa/1268745584


Export citation

References

  • C. C. Cowen, ``Composition operators on $H^2$,'' Journal of Operator Theory, vol. 9, no. 1, pp. 77--106, 1983.
  • E. A. Nordgren, ``Composition operators,'' Canadian Journal of Mathematics, vol. 20, pp. 442--449, 1968.
  • C. C. Cowen, ``Linear fractional composition operators on $H^2$,'' Integral Equations and Operator Theory, vol. 11, no. 2, pp. 151--160, 1988.
  • C. C. Cowen and T. L. Kriete III, ``Subnormality and composition operators on $H^2$,'' Journal of Functional Analysis, vol. 81, no. 2, pp. 298--319, 1988.
  • C. Hammond, ``On the norm of a composition operator with linear fractional symbol,'' Acta Universitatis Szegediensis, vol. 69, no. 3-4, pp. 813--829, 2003.
  • P. S. Bourdon, E. E. Fry, C. Hammond, and C. H. Spofford, ``Norms of linear-fractional composition operators,'' Transactions of the American Mathematical Society, vol. 356, no. 6, pp. 2459--2480, 2004.
  • E. L. Basor and D. Q. Retsek, ``Extremal non-compactness of composition operators with linear fractional symbol,'' Journal of Mathematical Analysis and Applications, vol. 322, no. 2, pp. 749--763, 2006.
  • C. Hammond, ``Zeros of hypergeometric functions and the norm of a composition operator,'' Computational Methods and Function Theory, vol. 6, no. 1, pp. 37--50, 2006.
  • S. Effinger-Dean, A. Johnson, J. Reed, and J. Shapiro, ``Norms of composition operators with rational symbol,'' Journal of Mathematical Analysis and Applications, vol. 324, no. 2, pp. 1062--1072, 2006.
  • S. Stević, ``Norm of weighted composition operators from Bloch space to $H_\mu ^\infty $ on the unit ball,'' Ars Combinatoria, vol. 88, pp. 125--127, 2008.
  • S. Stević, ``Norms of some operators from Bergman spaces to weighted and Bloch-type spaces,'' Utilitas Mathematica, vol. 76, pp. 59--64, 2008.
  • S. Stević, ``Norm and essential norm of composition followed by differentiation from $\alpha $-Bloch spaces to $H_\mu ^\infty $,'' Applied Mathematics and Computation, vol. 207, no. 1, pp. 225--229, 2009.
  • E. Wolf, ``Weighted composition operators between weighted Bergman spaces and weighted Banach spaces of holomorphic functions,'' Revista Matemática Complutense, vol. 21, no. 2, pp. 475--480, 2008.
  • F. Forelli, ``The isometries of $H^p$,'' Canadian Journal of Mathematics, vol. 16, pp. 721--728, 1964.
  • J. H. Clifford and M. G. Dabkowski, ``Singular values and Schmidt pairs of composition operators on the Hardy space,'' Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 183--196, 2005.
  • Z.-S. Fang and Z.-H. Zhou, ``Differences of composition operators on the space of bounded analytic functions in the polydisc,'' Abstract and Applied Analysis, vol. 2008, Article ID 983132, 10 pages, 2008.
  • X. Fu and X. Zhu, ``Weighted composition operators on some weighted spaces in the unit ball,'' Abstract and Applied Analysis, vol. 2008, Article ID 605807, 8 pages, 2008.
  • S. Li and S. Stević, ``Weighted composition operators from $H^\infty $ to the Bloch space on the polydisc,'' Abstract and Applied Analysis, vol. 2007, Article ID 48478, 13 pages, 2007.
  • S. Li and S. Stević, ``Weighted composition operators from Zygmund spaces into Bloch spaces,'' Applied Mathematics and Computation, vol. 206, no. 2, pp. 825--831, 2008.
  • L. Luo and S. Ueki, ``Weighted composition operators between weighted Bergman spaces and Hardy spaces on the unit ball of $C^n$,'' Journal of Mathematical Analysis and Applications, vol. 326, no. 1, pp. 88--100, 2007.
  • B. D. MacCluer and R. Zhao, ``Essential norms of weighted composition operators between Bloch-type spaces,'' The Rocky Mountain Journal of Mathematics, vol. 33, no. 4, pp. 1437--1458, 2003.
  • J. H. Shapiro and W. Smith, ``Hardy spaces that support no compact composition operators,'' Journal of Functional Analysis, vol. 205, no. 1, pp. 62--89, 2003.
  • S. Stević, ``Weighted composition operators between mixed norm spaces and $H_\alpha ^\infty $ spaces in the unit ball,'' Journal of Inequalities and Applications, vol. 2007, Article ID 28629, 9 pages, 2007.
  • S. Stević, ``Essential norms of weighted composition operators from the $\alpha $-Bloch space to a weighted-type space on the unit ball,'' Abstract and Applied Analysis, vol. 2008, Article ID 279691, 11 pages, 2008.
  • S.-I. Ueki and L. Luo, ``Compact weighted composition operators and multiplication operators between Hardy spaces,'' Abstract and Applied Analysis, vol. 2008, Article ID 196498, 12 pages, 2008.
  • X. Zhu, ``Weighted composition operators from $F(p,q,s)$spaces to $H_\mu ^\infty $ spaces,'' Abstract and Applied Analysis, vol. 2009, Article ID 290978, 14 pages, 2009.
  • C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, Fla, USA, 1995.
  • G. Gunatillake, Weighted composition operators, Ph.D. thesis, Purdue University, West Lafayette, Ind, USA, 2005.
  • C. Hammond, ``The norm of a composition operator with linear symbol acting on the Dirichlet space,'' Journal of Mathematical Analysis and Applications, vol. 303, no. 2, pp. 499--508, 2005.
  • P. S. Bourdon, D. Levi, S. K. Narayan, and J. H. Shapiro, ``Which linear-fractional composition operators are essentially normal?'' Journal of Mathematical Analysis and Applications, vol. 280, no. 1, pp. 30--53, 2003.