Skip to main content
SpringerLink
Log in

Multiplicity of Analytic Toeplitz Operators

  • Chapter
Toeplitz Operators and Spectral Function Theory

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 42))

Abstract

µ(T) of an operator T acting on a linear topological space x is defined to be the least cardinal number of Y⊂x such that

$$ Span\left( {{T^n}y:n \geqslant 0} \right) = x $$
((1.1))

. Any Y satisfying (1.1) is called a cyclic set for T.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aleksandrov, A.B.: Invariant subspaces of the shift operators, An axiomatic approach, Zapiski Nauch.Semin.L.O.M.I. 113 (1981), 7–26. (Russian)

    Google Scholar 

  2. Aleksandrov, A.B.; Essays on locally convex Hardy classes, Lecture Kotes in Math. 864 (1981), 1–90.

    Article  Google Scholar 

  3. Atzmon, A.: Multilinear mappings and estimates of multiplicity. Integral Equations and Operator Theory 10 (1987), 1–16.

    Article  Google Scholar 

  4. Baker I.N., Deadens, J.A. and Ullman, J.L.: A theorem on entire functions with applications to Toeplitz operators, Duke Math.J. 41 (1974). 739–745.

    Article  Google Scholar 

  5. Behnke, H. and Sommer, F.: Theorie der analytischen Funktionen einer komplexen Veränderlichen, 3 Aufl., Springer, Ber-lin-Heidelberg-New York, 1965.

    Book  Google Scholar 

  6. Bourdon, P.S.: Density of the polynomials in Bergman spaces, Pacific Math.J., 130 No.2 (1987), 215–222.

    Article  Google Scholar 

  7. Bram, J.: Subnormal operators, Duke Math.J., 22 (1955), 75–94.

    Article  Google Scholar 

  8. Clark, D.N.: On a similarity theory for rational Toeplitz operators, J.Reine Angew.Math. 320 (1980), 6–31.

    Google Scholar 

  9. Clark, D.N.: On Toeplitz operators with loops II, J.Operator Theory 7 (1982), 109–123.

    Google Scholar 

  10. Cowen, CO.: The commutant of an analytic Toeplitz operator II, Indiana Univ.Math.J. 29, No.1 (1980), 1–12.

    Article  Google Scholar 

  11. David, G.; Opérateurs intégraux singuliers sur certaines courbes du plan complexe, Ann.Sci.Ec.Norm.Sup., 4e série 17 (1984), 157–189.

    Google Scholar 

  12. Douady, A.: Le problème de modules pour sous-espaces analytiques compacts d’un espace analytique donné, Ann.Inst. Fourier 15, No.1 (1966), 1–94.

    Article  Google Scholar 

  13. Douglas, R.G., Shapiro, H.S., Shields, A.L.: Cyclic vectors and invariant subspaces for the backward shift operator, Ann. Inst.Fourier 20, .No.1 (1970), 37–76.

    Article  Google Scholar 

  14. Duren, E.M.: Theory of H spaces, Academic Press, New York-London, 1970.

    Google Scholar 

  15. Dyn’kin, E.M. : On the smoothness of Cauchy type integrals, Zapiski tfauch.Semin.L.O.M.I. 92 (1979), 115–133.

    Google Scholar 

  16. Forster, O.: Riemannsche flächen, Springer, Berlin— Heidelberg — Hew York, 1977.

    Book  Google Scholar 

  17. Gahov, F.D.: A case of the Riemann Problem for a system of n function pairs, Izvestia Akad. Nauk SSSR 14, .No.6 (1950), 549–568.

    Google Scholar 

  18. Gantmacher, P.R. : Matrizenrechung I, II, Deutscher Verlag der Wissenschaften, Berlin, 1958, 1966 (transl.from Russian).

    Google Scholar 

  19. Garnett, J.B.: Bounded analytic functions, Academic Press, New York, 1981.

    Google Scholar 

  20. Goluzin, G.M.: Geometric theory of functions of a complex variable, Amer.Math.Soc. R.I., Providence, 1969 (transi. from Russian).

    Google Scholar 

  21. Havin, V.P. : Boundary properties of the Cauchy integral and harmonic conjugate functions in the domains with rectifiable boundary, Mat.Sbornik 68, No.4 (1965), 499–517. (Russian).

    Google Scholar 

  22. Herrero D. : On multicyclic operators, Integral Equations and Operator Theory 94, No.1 (1978), 57–102.

    Article  Google Scholar 

  23. Herrero D. : On the multiplicities of the powers of a Banach space operator, Proc.Amer.Math.Soc. 94, .No. 2 (1985). 239–243.

    Article  Google Scholar 

  24. Herrero D.: The Fredholm structure of an n-multicyclic operator, Indiana Univ.Math.J. 36, No.3 (1987), 549–566.

    Article  Google Scholar 

  25. Herrero, D. and Rodman, L.; The multicyclic n-tuples of an n-multicyclic operator, and analytic structures on its spectrum, Indiana Univ.Math.J. 34, No.3 (1985), 619–629.

    Article  Google Scholar 

  26. Herrero, D. and Wogen, W. : On the multiplicity of T⊕T⊕... ⊕T Preprint, 1987.

    Google Scholar 

  27. Hitt, D.: Invariant subspaces of M of an annulus, Pacific J. of Math. 134, No.1 (1988) 101–120.

    Article  Google Scholar 

  28. Leiterer, J.; Holomorphic vector bundles and the Oka-Grauert principle, Sovrem.Probl.Math., Fund.Napr., 10 (1986), 75–123, ‘VI-NITI’, Moscow. (Russian)

    Google Scholar 

  29. Lin, V.Ya.: Holomorphic bundles and multivalued functions of elements of a Banach algebra, Funkc.Anal.i ego Pril. 7 No.2 (1973), 43–51. (Russian).

    Google Scholar 

  30. Nikol’skii, N.K. : Designs for calculating the spectral multiplicity of orthogonal sums, Zapiski Nauchn.Semin.L.O.M.I. 126 (1983), 150–158; English transi, in J.Soviet Math. 27, No.1 (1984).

    Google Scholar 

  31. Nikol’skii, N.K.: Ha-plitz operators: a survey of some recent results, Proc.Conf.Operators and .Function Theory, Lancaster 1984, 87–137, Reidel, 1984.

    Google Scholar 

  32. Nikol’skii, N.K.: Treatise on the shift operator, Springer-Verlag, 1986.

    Book  Google Scholar 

  33. Nikol’skii, N.K. and Vasyunin, V.I.: Control subspaces of minimal dimension and root vectors, Integral Equations and Operator Theory 6 (1983), 274–311.

    Article  Google Scholar 

  34. Nikol’skii, N.K. and Vasytuain, V.I.: Control subspaces of minimal dimension, unitary and model operators, J.Operator Theory 10 (1983), 307–330.

    Google Scholar 

  35. Nordgren, E.: Reducing subspaces of analytic Toeplitz operators, Duke Math.J., 34 (1967), 175–181.

    Article  Google Scholar 

  36. Privalov, I.I.: Randeigenshhaften Aaalytishher Functionen, Deutscher Verlag der Wiss., East Berlin, 1956.

    Google Scholar 

  37. Sarason, D.: The H spaces of an annulus, Mem .Amer .Math. Soc. 56 (1965), 1–78.

    Google Scholar 

  38. Smirnov, V.I.: Sur les formules de Cauchy et de Green et quelques problèmes qui s’y rattachent, Izvestia Akad.Nauk S.S.S-R, ser.mat. 7 (1932), 337–372.

    Google Scholar 

  39. Solomyak, B.M.: On the multiplicity of analytic Toeplitz operators, Dokl.Akad.Nauk S.S.S,R 286, No.6 (1986), 1308–1311; English transi.in Soviet Math.Dokl. 33 No.1 (1986).

    Google Scholar 

  40. Solomyak, B.M.: Caylic sets for analytic Toeplitz operators, Zapiski Nauchn.Semin.L.O.M.I. 157 (1987), 88–102. (Russian)

    Google Scholar 

  41. Solomyak, B.M.: Calculi, invariant subspaces and multiplicity for some classes of operators, Dissertation, Leningrad State University, 1986. (Russian)

    Google Scholar 

  42. Solomyak, B.M.: Multiplicity, calculi and multiplication operators, Sibirskii Matem.Jurnal. 29, No.2 (1988), 167–175. (Russian)

    Google Scholar 

  43. Stephenson, K.: Analytic functions of finite valence, with applications to Toeplitz operators, Michigan, Math.J. 32 No.1 (1985), 5–19.

    Article  Google Scholar 

  44. Stout, E.L.; Bounded holomorphic functions on finite Riemann surfaces, Trans.Amer.Math.Soc. 120, Ho 2 (Nov.1965), 255–285.

    Google Scholar 

  45. Sz.-Nagy, B. and Foias, G.: Compléments à l’étude des opérateurs de classe Co, Acta Sci.Math. 31, .No.3–4 (1970), 287–296.

    Google Scholar 

  46. Sz.-.Nagy, B. and Foias, C.: Compléments à l’étude des opérateurs de classe Co II, Acta Sci.Math. 23, No.1–2 (1972), 113–116.

    Google Scholar 

  47. Sz.-Nagy, B. and Foias, C.; Harmonic Analysis of Operators on Hubert space, North-Holland, 1970.

    Google Scholar 

  48. Thomson, J.E.; The commutant of a class of analytic Toeplitz operators, Amer.J.Math. 99 (1977), 522–529.

    Article  Google Scholar 

  49. Tolokonnikov, V.A.: Estimates in the Carleson Corona Theorem ideals of the algebra , a problem of Sz.-Nagy, Zapiski Nauchn.Semin.L.O.M.I. 113 (1981), 178–199. (Russian)

    Google Scholar 

  50. Tumarkin, G.C and Khavinson, S.Ya.: Classes of analytic functions on multiply-connected domains, Contemporary problems in the theory of functions of one complex variable, 45–77, Moscow, 1960, (Russian). French translation: Fonctions d’une variable complexe, Problèmes contemporains, 37–71, Gauthiers-Villars, Paris, 1962.

    Google Scholar 

  51. Tumarkin, G.C., and Khavinson, S.Ya.: Existence of a single-valued function in a given class with a given modulus of its boundary values in multiply connected domains, Izvestia Akad.-Nauk S.S.S.R. ser.Mat. 22 (1958, 543–562. (Russian)

    Google Scholar 

  52. Vasytuiin, V.E.: Multiplicity of contractions with finite defect indices, present volume.

    Google Scholar 

  53. Vasyunin, V.I., and Karaev, M.T.: The multiplicity of some contractions, Zapiski Nauchn.Semin. L.O.M.I. 157 (1987), 23–29. (Russian)

    Google Scholar 

  54. Vekua, N.P.: The Riemann problem with discontinuous coefficients for several functions, Soobsch.Akad.Nauk GSSR 2 No.9 (1950), 533–538. (Russian)

    Google Scholar 

  55. Volberg, A.L. : How to break through a given contour, L.O.M.I Preprints P-I-89, 1989 (Russian)

    Google Scholar 

  56. Volberg, A.L. and Solomyak, B.M.: The multiplicity of Toep-litz operators, weighted cocycles and the vector-valued Riemann problem. Funkc.Anal.i ego Pril. 21 No.3 (1987), 1–10. (Russian)

    Google Scholar 

  57. Volberg, A.L. and Solomyak, B.M.: Operator of multiplication by analytic matrix-valued function, present volume.

    Google Scholar 

  58. Wang, D.: Similarity of smooth Toeplitz operators, J.Opera-tor Theory 12 (1984), 319–348.

    Google Scholar 

  59. Warschawskii, S.E.: On conformai mapping of regions bounded by smooth curves, Proc.Amer.Math.Soc. 2 (1951), 254–261.

    Article  Google Scholar 

  60. Wermer, J.: Rings of analytic functions, Annals of Math. 67 No.3 (1958), 497–516.

    Article  Google Scholar 

  61. Wogen, W.R. : On some operators with cyclic vectors, Indiana Univ.Math.J. 27 No.1 (1978), 163–171.

    Article  Google Scholar 

  62. Yakubovich, D.V.: Similarity models of ïoeplita operatprs, Zapiski Nauchn.Semin.L.O.M.I. 157 (1987), 113–123. (Russian)

    Google Scholar 

  63. Yakubovich, D.V.: Rieraann surface models of Toeplitz operators, present volume.

    Google Scholar 

  64. Zinsmeister, M.: Domain de Lavrentiev, Publication Math. d’ORSAY, 85–03, 1985.

    Google Scholar 

  65. Dyn’kin E.M. Methods of the theory of singular integrals. in Commutative Harmonic Analysis I. Encyclopaedia of Mathematical Sciences. Springer-Verlag, 1989.

    Google Scholar 

Download references

Authors
  1. B. M. Solomyak
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. L. Volberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

N. K. Nikolskii

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Springer Basel AG

About this chapter

Cite this chapter

Solomyak, B.M., Volberg, A.L. (1989). Multiplicity of Analytic Toeplitz Operators. In: Nikolskii, N.K. (eds) Toeplitz Operators and Spectral Function Theory. Operator Theory: Advances and Applications, vol 42. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5587-7_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-5587-7_3

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-5589-1

  • Online ISBN: 978-3-0348-5587-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation