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Spectra of composition operators on the bloch and bergman spaces

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Abstract

Whenϕ is an analytic map of the unit diskU into itself, andX is a Banach space of analytic functions onU, define the composition operatorC ϕ byC ϕ (f)=f o ϕ, forf∈X. In this paper we show how to use the Calderón theory of complex interpolation to obtain information on the spectrum ofC ϕ (under suitable hypotheses onϕ) acting on the Bloch spaceB and BMOA, the space of analytic functions in BMO. To do this we first obtain some results on the essential spectral radius and spectrum ofC ϕ on the Bergman spacesA pand Hardy spacesH p,spaces which are connected toB and BMOA by the interpolation relationships [A 1,B] t =A pand [H 1,BMOA] t =H pfor 1=p(1−t).

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References

  1. J. Arazy, S. Fisher and J. Peetre,Möbius invariant function spaces, Journal für die reine und angewandte Mathematik363 (1985), 110–145.

    MATH  MathSciNet  Google Scholar 

  2. S. Axler,Bergman spaces and their operators, inSurveys of Some Recent Results in Operator Theory Vol. 1 (J. B. Conway and B. B Morrel, eds.), Pitman Research Notes in Mathematics171 (1988), 1–50.

    MathSciNet  Google Scholar 

  3. J. Bergh and J. Löfström,Interpolation Spaces, Springer-Verlag, Berlin, 1976.

    MATH  Google Scholar 

  4. P. Bourdon,Essential angular derivatives and maximum growth of Koenigs eigen-functions, Journal of Functional Analysis160 (1998), 561–580.

    Article  MATH  MathSciNet  Google Scholar 

  5. P. Bourdon, J. Cima and A. Matheson,Compact composition operators on BMOA, Transactions of the American Mathematical Society351 (1999), 2183–2196.

    Article  MATH  MathSciNet  Google Scholar 

  6. P. Bourdon and J. Shapiro,Mean growth of Koenigs eigenfunctions, Journal of the American Mathematical Society10 (1997), 299–325.

    Article  MATH  MathSciNet  Google Scholar 

  7. A. Calderón,Intermediate spaces and interpolation, the complex method, Studia Mathematica24 (1964), 113–190.

    MATH  MathSciNet  Google Scholar 

  8. C. Cowen and B. MacCluer,Spectra of some composition operators, Journal of Functional Analysis125 (1994), 223–251.

    Article  MATH  MathSciNet  Google Scholar 

  9. C. Cowen and B. MacCluer,Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1995.

    MATH  Google Scholar 

  10. M. Cwikel, M. Milman and Y. Sagher,Complex interpolation of some quasi-Banach spaces, Journal of Functional Analysis65 (1986), 339–347.

    Article  MATH  MathSciNet  Google Scholar 

  11. R. Donaway,Norm and essential norm estimates of composition operators on Besov type spaces, Ph.D. Dissertation, University of Virginia, 1999.

  12. H. Kamowitz,The spectra of endomorphisms of the disc algebra, Pacific Journal of Mathematics46 (1973), 433–440.

    MATH  MathSciNet  Google Scholar 

  13. K. Madigan and A. Matheson,Compact composition operators on the Bloch space, Transactions of the American Mathematical Society347 (1995), 2679–2687.

    Article  MATH  MathSciNet  Google Scholar 

  14. A. Montes-Rodríguez,The essential norm of a composition operator on the Bloch spaces, Pacific Journal of Mathematics188 (1999), 339–351.

    Article  MATH  MathSciNet  Google Scholar 

  15. P. Poggi-Corradini,The Hardy class of Koenigs maps, The Michigan Mathematical Journal44 (1997), 495–507.

    Article  MATH  MathSciNet  Google Scholar 

  16. P. Poggi-Corradini,The essential norm of composition operators revisited, inStudies on Composition Operators (Laramie, WY, 1996), Contemporary Mathematics, Vol. 213, American Mathematical Society, Providence, RI, 1998, pp. 167–173.

    Google Scholar 

  17. W. Rudin,Function Theory in the Unit Ball of ℂ n,Springer-Verlag, Berlin, 1980.

    Google Scholar 

  18. K. Saxe,Compactness-like properties preserved by complex interpolation, Arkiv för Matematik35 (1997), 353–362.

    Article  MATH  MathSciNet  Google Scholar 

  19. J. Shapiro,The essential norm of a composition operator, Annals of Mathematics125 (1987), 375–404.

    Article  MathSciNet  Google Scholar 

  20. W. Smith,Composition operators between Bergman and Hardy spaces, Transactions of the American Mathematical Society348 (1996), 2331–2348.

    Article  MATH  MathSciNet  Google Scholar 

  21. M. Tjani,Compact composition operators on some Möbius invariant Banach spaces, Ph.D. Dissertation, Michigan State University, 1996.

  22. D. Vukotić, A sharp estimate forA p a functions in ℂn, Proceedings of the American Mathematical Society117 (1993), 753–756.

    Article  MathSciNet  MATH  Google Scholar 

  23. T. Wolff,A note on interpolation spaces, inHarmonic Analysis (Minneapolis, MN, 1981), Lecture Notes in Mathematics908, Springer-Verlag, Berlin, 1982, pp. 199–204.

    Chapter  Google Scholar 

  24. L. Zheng, Essential norm and spectrum of composition operators onH ∞, preprint.

  25. K. Zhu,Operator Theory in Function Spaces, Marcel Dekker, New York, 1990.

    MATH  Google Scholar 

  26. K. Zhu and T. MacGregor,Coefficient multipliers between Bergman and Hardy spaces, Mathematika42 (1995), 413–426.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Kerchof Hall, University of Virginia, 22904, Charlottesville, VA, USA

    Barbara MacCluer

  2. Department of Mathematics, Macalester College, 1600 Grand Avenue, 55105, St. Paul, MN, USA

    Karen Saxe

Authors
  1. Barbara MacCluer
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  2. Karen Saxe
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MacCluer, B., Saxe, K. Spectra of composition operators on the bloch and bergman spaces. Isr. J. Math. 128, 325–354 (2002). https://doi.org/10.1007/BF02785430

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