International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2007 / Article

Research Article | Open Access

Volume 2007 |Article ID 092070 | https://doi.org/10.1155/2007/92070

J. S. Manhas, "Composition Operators and Multiplication Operators on Weighted Spaces of Analytic Functions", International Journal of Mathematics and Mathematical Sciences, vol. 2007, Article ID 092070, 21 pages, 2007. https://doi.org/10.1155/2007/92070

Composition Operators and Multiplication Operators on Weighted Spaces of Analytic Functions

Academic Editor: Marianna A. Shubov
Received17 Jun 2006
Accepted26 Apr 2007
Published23 Jan 2008

Abstract

Let V be an arbitrary system of weights on an open connected subset G of N(N1) and let B(E) be the Banach algebra of all bounded linear operators on a Banach space E. Let HVb(G,E) and HV0(G,E) be the weighted locally convex spaces of vector-valued analytic functions. In this survey, we present a development of the theory of multiplication operators and composition operators from classical spaces of analytic functions H(G) to the weighted spaces of analytic functions HVb(G,E) and HV0(G,E).

References

  1. J. C. Evard and F. Jafari, “On semigroups of operators on Hardy spaces,” preprint, 1995. View at: Google Scholar
  2. A. G. Siskakis, “Weighted composition semigroups on Hardy spaces,” Linear Algebra and Its Applications, vol. 84, pp. 359–371, 1986. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  3. A. G. Siskakis, “Semigroups of composition operators on spaces of analytic functions, a review,” in Studies on Composition Operators (Laramie, WY, 1996), vol. 213 of Contemporary Mathematics, pp. 229–252, American Mathematical Society, Providence, RI, USA, 1998. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  4. K. de Leeuw, W. Rudin, and J. Wermer, “The isometries of some function spaces,” Proceedings of the American Mathematical Society, vol. 11, no. 5, pp. 694–698, 1960. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  5. M. Nagasawa, “Isomorphisms between commutative Banach algebras with an application to rings of analytic functions,” Kōdai Mathematical Seminar Reports, vol. 11, pp. 182–188, 1959. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  6. F. Forelli, “The isometries of Hp,” Canadian Journal of Mathematics, vol. 16, pp. 721–728, 1964. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  7. M. Cambern and K. Jarosz, “The isometries of H1,” Proceedings of the American Mathematical Society, vol. 107, no. 1, pp. 205–214, 1989. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  8. C. J. Kolaski, “Surjective isometries of weighted Bergman spaces,” Proceedings of the American Mathematical Society, vol. 105, no. 3, pp. 652–657, 1989. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  9. T. Mazur, “Canonical isometry on weighted Bergman spaces,” Pacific Journal of Mathematics, vol. 136, no. 2, pp. 303–310, 1989. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  10. P.-K. Lin, “The isometries of H(E),” Pacific Journal of Mathematics, vol. 143, no. 1, pp. 69–77, 1990. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  11. W. Arveson, “Subalgebras of C-algebras—III: multivariable operator theory,” Acta Mathematica, vol. 181, no. 2, pp. 159–228, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  12. K. R. M. Attele, “Analytic multipliers of Bergman spaces,” The Michigan Mathematical Journal, vol. 31, no. 3, pp. 307–319, 1984. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  13. S. Axler, “Multiplication operators on Bergman spaces,” Journal für die reine und angewandte Mathematik, vol. 336, pp. 26–44, 1982. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  14. S. Axler, “Zero multipliers of Bergman spaces,” Canadian Mathematical Bulletin, vol. 28, no. 2, pp. 237–242, 1985. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  15. S. Axler, “The Bergman space, the Bloch space, and commutators of multiplication operators,” Duke Mathematical Journal, vol. 53, no. 2, pp. 315–332, 1986. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  16. S. Axler and A. L. Shields, “Univalent multipliers of the Dirichlet space,” The Michigan Mathematical Journal, vol. 32, no. 1, pp. 65–80, 1985. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  17. H. Bercovici, “The algebra of multiplication operators on Bergman spaces,” Archiv der Mathematik, vol. 48, no. 2, pp. 165–174, 1987. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  18. J. Eschmeier, “Multiplication operators on Bergman spaces are reflexive,” in Linear Operators in Function Spaces (Timişoara, 1988), H. Helson, B. Sz-Nagy, F.-H. Vasilescu, and Gr. Arsene, Eds., vol. 43 of Operator Theory Adv. Appl., pp. 165–184, Birkhäuser, Basel, Switzerland, 1990. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  19. D. H. Luecking, “Multipliers of Bergman spaces into Lebesgue spaces,” Proceedings of the Edinburgh Mathematical Society, vol. 29, no. 1, pp. 125–131, 1986. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  20. D. Vukotić, “Pointwise multiplication operators between Bergman spaces on simply connected domains,” Indiana University Mathematics Journal, vol. 48, no. 3, pp. 793–803, 1999. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  21. K. Zhu, “A trace formula for multiplication operators on invariant subspaces of the Bergman space,” Integral Equations and Operator Theory, vol. 40, no. 2, pp. 244–255, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  22. D. M. Campbell and R. J. Leach, “A survey of Hp multipliers as related to classical function theory,” Complex Variables, vol. 3, no. 1–3, pp. 85–111, 1984. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  23. N. S. Feldman, “Pointwise multipliers from the Hardy space to the Bergman space,” Illinois Journal of Mathematics, vol. 43, no. 2, pp. 211–221, 1999. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  24. S. Ohno and H. Takagi, “Some properties of weighted composition operators on algebras of analytic functions,” Journal of Nonlinear and Convex Analysis, vol. 2, no. 3, pp. 369–380, 2001. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  25. J. Arazy, Multipliers of Bloch Functions, vol. 54, University of Haifa Mathematics, Haifa, Israel, 1982. View at: Google Scholar
  26. L. Brown and A. L. Shields, “Multipliers and cyclic vectors in the Bloch space,” The Michigan Mathematical Journal, vol. 38, no. 1, pp. 141–146, 1991. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  27. D. A. Stegenga, “Multipliers of the Dirichlet space,” Illinois Journal of Mathematics, vol. 24, no. 1, pp. 113–139, 1980. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  28. J. M. Ortega and J. Fàbrega, “Pointwise multipliers and corona type decomposition in BMOA,” Annales de l'Institut Fourier, vol. 46, no. 1, pp. 111–137, 1996. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  29. H. Jarchow, V. Montesinos, K. J. Wirths, and J. Xiao, “Duality for some large spaces of analytic functions,” Proceedings of the Edinburgh Mathematical Society, vol. 44, no. 3, pp. 571–583, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  30. N. Yanagihara, “Multipliers and linear functionals for the class N+,” Transactions of the American Mathematical Society, vol. 180, pp. 449–461, 1973. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  31. J. Bonet, P. Domański, and M. Lindström, “Pointwise multiplication operators on weighted Banach spaces of analytic functions,” Studia Mathematica, vol. 137, no. 2, pp. 177–194, 1999. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  32. J. Bonet, P. Domański, M. Lindström, and J. Taskinen, “Composition operators between weighted Banach spaces of analytic functions,” Journal of Australian Mathematical Society, vol. 64, no. 1, pp. 101–118, 1998. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  33. J. Bonet, P. Domański, and M. Lindström, “Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions,” Canadian Mathematical Bulletin, vol. 42, no. 2, pp. 139–148, 1999. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  34. J. Bonet, P. Domański, and M. Lindström, “Weakly compact composition operators on analytic vector-valued function spaces,” Annales Academiæ Scientiarium Fennicæ. Mathematica, vol. 26, no. 1, pp. 233–248, 2001. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  35. M. D. Contreras and A. G. Hernández-Díaz, “Weighted composition operators in weighted Banach spaces of analytic functions,” Journal of Australian Mathematical Society, vol. 69, no. 1, pp. 41–60, 2000. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  36. A. L. Shields and D. L. Williams, “Bonded projections, duality, and multipliers in spaces of analytic functions,” Transactions of the American Mathematical Society, vol. 162, pp. 287–302, 1971. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  37. A. L. Shields and D. L. Williams, “Bounded projections, duality, and multipliers in spaces of harmonic functions,” Journal für die reine und angewandte Mathematik, vol. 299/300, pp. 256–279, 1978. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  38. M. D. Contreras and A. G. Hernández-Díaz, “Weighted composition operators on Hardy spaces,” Journal of Mathematical Analysis and Applications, vol. 263, no. 1, pp. 224–233, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  39. G. Mirzakarimi and K. Seddighi, “Weighted composition operators on Bergman and Dirichlet spaces,” Georgian Mathematical Journal, vol. 4, no. 4, pp. 373–383, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  40. 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. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  41. S. Ohno, “Weighted composition operators between H and the Bloch space,” Taiwanese Journal of Mathematics, vol. 5, no. 3, pp. 555–563, 2001. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  42. S. Ohno and R. Zhao, “Weighted composition operators on the Bloch space,” Bulletin of the Australian Mathematical Society, vol. 63, no. 2, pp. 177–185, 2001. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  43. S. Ohno, K. Stroethoff, and R. Zhao, “Weighted composition operators between Bloch-type spaces,” The Rocky Mountain Journal of Mathematics, vol. 33, no. 1, pp. 191–215, 2003. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  44. A. Montes-Rodríguez, “Weighted composition operators on weighted Banach spaces of analytic functions,” Journal of the London Mathematical Society, vol. 61, no. 3, pp. 872–884, 2000. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  45. F. Jafari, T. Tonev, E. Toneva, and K. Yale, “Holomorphic flows, cocycles, and coboundaries,” The Michigan Mathematical Journal, vol. 44, no. 2, pp. 239–253, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  46. 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. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  47. J. H. Shapiro, Composition Operators and Classical Function Theory, Universitext: Tracts in Mathematics, Springer, New York, NY, USA, 1993. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  48. R. K. Singh and J. S. Manhas, Composition Operators on Function Spaces, vol. 179 of North-Holland Mathematics Studies, North-Holland, Amsterdam, The Netherlands, 1993. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  49. K. D. Bierstedt, J. Bonet, and A. Galbis, “Weighted spaces of holomorphic functions on balanced domains,” The Michigan Mathematical Journal, vol. 40, no. 2, pp. 271–297, 1993. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  50. K. D. Bierstedt, J. Bonet, and J. Taskinen, “Associated weights and spaces of holomorphic functions,” Studia Mathematica, vol. 127, no. 2, pp. 137–168, 1998. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  51. K. D. Bierstedt and W. H. Summers, “Biduals of weighted Banach spaces of analytic functions,” Journal of Australian Mathematical Society, vol. 54, no. 1, pp. 70–79, 1993. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  52. A. Galbis, “Weighted Banach spaces of entire functions,” Archiv der Mathematik, vol. 62, no. 1, pp. 58–64, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  53. D. García, M. Maestre, and P. Sevilla-Peris, “Composition operators between weighted spaces of holomorphic functions on Banach spaces,” Annales Academiæ Scientiarium Fennicæ. Mathematica, vol. 29, no. 1, pp. 81–98, 2004. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  54. W. Lusky, “On weighted spaces of harmonic and holomorphic functions,” Journal of the London Mathematical Society, vol. 51, no. 2, pp. 309–320, 1995. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  55. J. Burbea and P. Masani, Banach and Hilbert Spaces of Vector-Valued Functions, vol. 90 of Research Notes in Mathematics, Pitman, Boston, Mass, USA, 1984. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  56. J. B. Garnett, Bounded Analytic Functions, vol. 96 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1981. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  57. A. Grothendieck, Topological Vector Spaces, Gordon and Breach Science, New York, NY, USA, 1992. View at: Google Scholar | Zentralblatt MATH
  58. W. Rudin, Functional Analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill, New York, NY, USA, 1973. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  59. L. A. Rubel and A. L. Shields, “The second duals of certain spaces of analytic functions,” Journal of Australian Mathematical Society, vol. 11, pp. 276–280, 1970. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  60. J. S. Manhas, “Multiplication operators on weighted locally convex spaces of vector-valued analytic functions,” Southeast Asian Bulletin of Mathematics, vol. 27, no. 4, pp. 649–660, 2003. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  61. K. Cichoń and K. Seip, “Weighted holomorphic spaces with trivial closed range multiplication operators,” Proceedings of the American Mathematical Society, vol. 131, no. 1, pp. 201–207, 2003. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  62. R. K. Singh and J. S. Manhas, “Invertible composition operators on weighted function spaces,” Acta Scientiarum Mathematicarum, vol. 59, no. 3-4, pp. 489–501, 1994. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  63. A. Pietsch, Operator Ideals, vol. 16 of Mathematical Monographs, VEB Deutscher Verlag der Wissenschaften, Berlin, Germany, 1978. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  64. P. Rueda, “On the Banach-Dieudonné theorem for spaces of holomorphic functions,” Quaestiones Mathematicae, vol. 19, no. 1-2, pp. 341–352, 1996. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  65. D. García, M. Maestre, and P. Rueda, “Weighted spaces of holomorphic functions on Banach spaces,” Studia Mathematica, vol. 138, no. 1, pp. 1–24, 2000. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  66. J. S. Manhas, “Homomorphisms and composition operators on weighted spaces of analytic functions,” preprint, 2006. View at: Google Scholar
  67. R. Aron, P. Galindo, and M. Lindström, “Compact homomorphisms between algebras of analytic functions,” Studia Mathematica, vol. 123, no. 3, pp. 235–247, 1997. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  68. P. Galindo, M. Lindström, and R. Ryan, “Weakly compact composition operators between algebras of bounded analytic functions,” Proceedings of the American Mathematical Society, vol. 128, no. 1, pp. 149–155, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  69. J. Bonet and M. Friz, “Weakly compact composition operators on locally convex spaces,” Mathematische Nachrichten, vol. 245, no. 1, pp. 26–44, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

Copyright © 2007 J. S. Manhas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

 PDF Download Citation Citation
 Order printed copiesOrder
Views220
Downloads711
Citations

Related articles

Article of the Year Award: Outstanding research contributions of 2020, as selected by our Chief Editors. Read the winning articles.