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ArticleOriginal scientific text

Title

Pointwise multiplication operators on weighted Banach spaces of analytic functions

Authors 1, 2, 3

Affiliations

  1. Departamento de Matemática Aplicada, Universidad Politécnica de Valencia, E-46071 Valencia, Spain
  2. Institute of Mathematics (Poznań branch), Polish Academy of Sciences, Matejki 48/49, 60-769 Poznań, Poland
  3. Department of Mathematics, Åbo Akademi University, FIN-20500 Åbo, Finland

Abstract

For a wide class of weights we find the approximative point spectrum and the essential spectrum of the pointwise multiplication operator Mφ, Mφ(f)=φf, on the weighted Banach spaces of analytic functions on the disc with the sup-norm. Thus we characterize when M′φ is Fredholm or is an into isomorphism. We also study cyclic phenomena for the adjoint map M′φ.

Keywords

ENG
weighted Banach spaces of analytic functions, pointwise multiplication operator, essential norm, closed range, approximative point spectrum, maximal ideal space of H∞, Shilov boundary, Gleason part, hypercyclic operator, chaotic operator

Bibliography

  1. [A] S. Axler, Multiplication operators on Bergman spaces, J. Reine Angew. Math. 336 (1982), 26-44.
  2. [BO] B. Berndtsson and J. Ortega-Cerdà, On interpolation and sampling in Hilbert spaces of analytic functions, ibid. 464 (1995), 109-128.
  3. [BBT] K. D. Bierstedt, J. Bonet and J. Taskinen, Associated weights and spaces of holomorphic functions, Studia Math. 127 (1998), 70-79.
  4. [BS] K. D. Bierstedt and W. H. Summers, Biduals of weighted Banach spaces of analytic functions, J. Austral. Math. Soc. Ser. A 54 (1993), 70-79.
  5. [BDL] 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.
  6. [BDLT] J. Bonet, P. Domański, M. Lindström and J. Taskinen, Composition operators between weighted Banach spaces of analytic functions, J. Austral. Math. Soc. 64 (1998), 101-118.
  7. [B] P. S. Bourdon, Invariant manifolds of hypercyclic vectors, Proc. Amer. Math. Soc. 118 (1993), 845-847.
  8. [ChD] M. D. Choi and C. Davis, The spectral mapping theorem for joint approximative point spectrum, Bull. Amer. Math. Soc. 80 (1974), 317-321.
  9. [C] J. B. Conway, A Course in Functional Analysis, Springer, Berlin, 1990.
  10. [DL] P. Domański and M. Lindström, Sets of interpolation and sampling for weighted Banach spaces of holomorphic functions, preprint, 1998.
  11. [G] J. B. Garnett, Bounded Analytic Functions, Academic Press, 1981.
  12. [GS] G. Godefroy and J. H. Shapiro, Operators with dense, invariant, cyclic vector manifold, J. Funct. Anal. 98 (1991), 229-269.
  13. [Go] P. Gorkin, Functions not vanishing on trivial Gleason parts of Douglas algebras, Proc. Amer. Math. Soc. 104 (1988), 1086-1090.
  14. [H] K. Hoffman, Bounded analytic functions and Gleason parts, Ann. of Math. 86 (1967), 74-111.
  15. [KL] A. Kerr-Lawson, Some lemmas on interpolating Blaschke products and a correction, Canad. J. Math. 21 (1969), 531-534.
  16. [L1] W. Lusky, On the structure of Hv0(D) and hv0(D), Math. Nachr. 159 (1992), 279-289.
  17. [L2] W. Lusky, On weighted spaces of harmonic and holomorphic functions, J. London Math. Soc. (2) 51 (1995), 309-320.
  18. [MS] G. McDonald and C. Sundberg, Toeplitz operators on the disc, Indiana Univ. Math. J. 28 (1979), 595-611.
  19. [N] A. Nicolau, Finite products of interpolating Blaschke products, J. London Math. Soc. 50 (1994), 520-531.
  20. [RS] L. A. Rubel and A. L. Shields, The second duals of certain spaces of analytic functions, J. Austral. Math. Soc. 11 (1970), 276-280.
  21. [R1] W. Rudin, Functional Analysis, McGraw-Hill, 1974.
  22. [R2] W. Rudin, Real and Complex Analysis, McGraw-Hill, 1974.
  23. [S] K. Seip, Beurling type density theorems in the unit disk, Invent. Math. 113 (1993), 21-39.
  24. [S1] K. Seip, On Korenblum's density condition for the zero sequences of A-α, J. Anal. Math. 67 (1995), 307-322.
  25. [S2] K. Seip, Developments from nonharmonic Fourier series, in: Proc. ICM 1998, Vol. II, 713-722.
  26. [Sh] J. O. Shapiro, Composition Operators and Classical Function Theory, Springer, 1993.
  27. [SW] A. L. Shields and D. L. Williams, Bounded projections and the growth of harmonic conjugates in the disk, Michigan Math. J. 29 (1982), 3-25.
  28. [SZ] Z. Słodkowski and W. Żelazko, On joint spectra of commuting families of operators, Studia Math. 50 (1974), 127-148.
  29. [V] D. Vukotić, Pointwise multiplication operators between Bergman spaces on simply connected domains, Indiana Univ. Math. J., to appear.
  30. [Z] W. Żelazko, An axiomatic approach to joint spectra I, Studia Math. 64 (1979), 249-261.
Studia Mathematica
1999/137/2
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Pages:
177-194
Main language of publication
English
Received
1999-01-07
Accepted
1999-08-23
Published
1999
Exact and natural sciences