SpringerLink
Log in

Contractions T for which A is a projection

  • Published:
Acta Scientiarum Mathematicarum Aims and scope Submit manuscript

Abstract

If T is a Hilbert space contraction, then

, where A is a nonnegative contraction. The strong limit A is a projection if and only T = G ⊕ V, where G is a strongly stable contraction and V is an isometry. This article is an expository paper on the class of contractions T for which A is a projection. After surveying such a class, it is shown that it is quite a large class. Indeed, it includes (i) all contractions whose adjoint has property PF, and also (ii) all contractions whose intersection of the continuous spectrum of its completely nonunitary direct summand with the unit circle has Lebesgue measure zero. Some new questions are investigated as well. For instance, is A a projection for every biquasitriangular contraction T? If so, then every contraction not in class C00 has a nontrivial invariant subspace.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H. Bercovici, Notes on invariant subspaces, Bull. Amer. Math. Soc., 23 (1990), 1–36.

    MathSciNet  MATH  Google Scholar 

  2. C. A. Berger and B. I. Shaw, Self-commutators of multicyclic hyponormal operators are always trace class, Bull. Amer. Math. Soc., 79 (1973), 1193–1199.

    MathSciNet  MATH  Google Scholar 

  3. C. A. Berger and B. I. Shaw, Intertwining, analytic structures, and the trace norm estimate, Proceedings of a Conference on Operator Theory, Halifax, 1973, Lecture Notes in Math. Vol. 345, Springer, Berlin, 1973, 1–6.

    Google Scholar 

  4. S. W. Brown, Hyponormal operators with thick spectra have invariant subspaces, Ann. Math., 125 (1987), 93–103.

    MathSciNet  MATH  Google Scholar 

  5. S. W. Brown, B. Chevreau and C. Pearcy, On the structure of contractions operators II, J. Functional Anal., 76 (1988), 30–55.

    MathSciNet  MATH  Google Scholar 

  6. J. B. Conway, The Theory of Subnormal Operators, Mathematical Surveys and Monographs Vol. 36, Amer. Math. Soc., Providence, 1991.

    MATH  Google Scholar 

  7. B. P. Duggal, On generalised Putnam-Fuglede theorems, Monatsh. Math., 107 (1989), 309–332.

    MathSciNet  MATH  Google Scholar 

  8. B. P. Duggal, On unitary parts of contractions, Indian J. Pure Appl. Math., 25 (1994), 1243–1247.

    MathSciNet  MATH  Google Scholar 

  9. B. P. Duggal, On characterising contractions with C10 pure part, Integral Equations Operator Theory, 27 (1997), 314–323.

    MathSciNet  MATH  Google Scholar 

  10. B. P. Duggal, S. V. Djordjević and C. S. Kubrusly, Hereditarily normaloid contractions, Acta Sci. Math. (Szeged), 71 (2005), 337–352.

    MathSciNet  MATH  Google Scholar 

  11. B. P. Duggal and C. S. Kubrusly, Paranormal contractions have property PF, Far East J. of Math. Sci., 14 (2004), 237–249.

    MathSciNet  MATH  Google Scholar 

  12. E. Durszt, Contractions as restricted shifts, Acta Sci. Math. (Szeged), 48 (1985), 129–134.

    MathSciNet  MATH  Google Scholar 

  13. G. P. Gehér, Positive operators arising asymptotically from contractions, Acta Sci. Math. (Szeged), 79 (2013), 273–287.

    MathSciNet  MATH  Google Scholar 

  14. E. Goya and T. Saitô, On intertwining by an operator having a dense range, Tôhoku Math. J., 33 (1981), 127–131.

    MathSciNet  MATH  Google Scholar 

  15. P. A. Fillmore, Notes on Operator Theory, Van Nostrand, New York, 1970.

    MATH  Google Scholar 

  16. P. R. Halmos, Quasitriangular operators, Acta Sci. Math. (Szeged), 29 (1968), 283–293.

    MathSciNet  MATH  Google Scholar 

  17. P. R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc., 76 (1970), 887–933.

    MathSciNet  MATH  Google Scholar 

  18. P. R. Halmos, A Hilbert Space Problem Book, Springer, New York, 1982.

    MATH  Google Scholar 

  19. P. R. Halmos and L. J. Wallen, Powers of partial isometries, J. Math. Mech., 19 (1970), 657–663.

    MathSciNet  MATH  Google Scholar 

  20. D. A. Herrero, Approximation of Hilbert Space Operators - Volume I, Longman, Harlow, 1989.

    Google Scholar 

  21. L. Kérchy, Isometric asymptotes of power bounded operators, Indiana Univ. Math. J., 38 (1989), 173–188.

    MathSciNet  MATH  Google Scholar 

  22. L. Kérchy, Unitary asymptotes of Hilbert space operators, Functional Analysis and Operator Theory, Warsaw, 1992, Banach Center Publ. Vol. 30, Polish Acad. Sci., Warsaw, 1994, 191–201.

    Google Scholar 

  23. L. Kérchy, Operators with regular norm-sequences, Acta Sci. Math. (Szeged), 63 (1997), 571–605.

    MathSciNet  MATH  Google Scholar 

  24. L. Kérchy, Generalized Toeplitz operators, Acta Sci. Math. (Szeged), 68 (2002), 373–400.

    MathSciNet  MATH  Google Scholar 

  25. L. Kérchy, Generalized Toeplitz operators associated with operators of regular norm-sequences, Semigroup of Operators: Theory and Applications, Optimization Software, Los Angeles, 2002, 119–131.

    Google Scholar 

  26. M. Kosiek and L. Suciu, Decompositions and asymptotic limit for bicontraction, Ann. Polon. Math., 105 (2012), 43–64.

    MathSciNet  MATH  Google Scholar 

  27. C. S. Kubrusly, Equivalent invariant subspace problems, J. Operator Theory, 38 (1997), 323–328.

    MathSciNet  MATH  Google Scholar 

  28. C. S. Kubrusly, An Introduction to Models and Decompositions in Operator Theory, Birkhäuser, Boston, 1997.

    MATH  Google Scholar 

  29. C. S. Kubrusly, Hilbert Space Operators: A Problem Solving Approach, Birkhäuser, Boston, 2003.

    MATH  Google Scholar 

  30. C. S. Kubrusly, Spectral Theory of Operators on Hilbert Spaces, Birkhäuser-Springer, New York, 2012.

    MATH  Google Scholar 

  31. C. S. Kubrusly and B. P. Duggal, Contractions with C·0 direct summands, Adv. Math. Sci. Appl., 11 (2001), 593–601.

    MathSciNet  MATH  Google Scholar 

  32. C. S. Kubrusly and N. Levan, Proper contractions and invariant subspaces, Int. J. Math. Math. Sci., 28 (2001), 223–230.

    MathSciNet  MATH  Google Scholar 

  33. C. S. Kubrusly and P. C. M. Vieira, Strong stability for cohyponormal operators, J. Operator Theory, 31 (1994), 123–127.

    MathSciNet  MATH  Google Scholar 

  34. C. S. Kubrusly, P. C. M. Vieira and D. O. Pinto, A decomposition for a class of contractions, Adv. Math. Sci. Appl., 6 (1996), 523–530.

    MathSciNet  MATH  Google Scholar 

  35. H. Langer, Ein Zerspaltungssatz für Operatoren im Hilbertraum, Acta Math. Acad. Sci. Hungar., 12 (1961), 441–445.

    MathSciNet  MATH  Google Scholar 

  36. N. Levan, Canonical decompositions of completely nonunitary contractions, J. Math. Anal. Appl., 101 (1984), 514–526.

    MathSciNet  MATH  Google Scholar 

  37. M. Martin and M. Putinar, Lectures on Hyponormal Operators, Birkhäuser, Basel, 1989.

    MATH  Google Scholar 

  38. K. Okubo, The unitary part of paranormal operators, Hokkaido Math. J., 6 (1977), 273–275.

    MathSciNet  MATH  Google Scholar 

  39. P. Pagacz, On Wold-type decomposition, Linear Algebra Appl., 436 (2012), 3065–3071.

    MathSciNet  MATH  Google Scholar 

  40. C. M. Pearcy, Some Recent Developments in Operator Theory, CBMS Regional Conference Series in Mathematics No. 36, Amer. Math. Soc., Providence, 1978.

    MATH  Google Scholar 

  41. V. Pták and P. Vrbová, An abstract model for compressions, Èasopis Pìst. Mat., 113 (1988), 252–266.

    MathSciNet  MATH  Google Scholar 

  42. C. R. Putnam, An inequality for the area of hyponormal spectra, Math. Z., 116 (1970), 323–330.

    MathSciNet  MATH  Google Scholar 

  43. C. R. Putnam, Spectra of polar factors of hyponormal operators, Trans. Amer. Math. Soc., 188 (1974), 419–428.

    MathSciNet  MATH  Google Scholar 

  44. C. R. Putnam, Hyponormal contractions and strong power convergence, Pacific J. Math., 57 (1975), 531–538.

    MathSciNet  MATH  Google Scholar 

  45. H. Radjavi and P. Rosenthal, Invariant Subspaces, Springer, Berlin, 1973; 2nd edn. Dover, New York, 2003.

    MATH  Google Scholar 

  46. H. Radjavi and P. Rosenthal, The invariant subspace problem, Math. Intelligencer, 4 (1982), 33–37.

    MathSciNet  MATH  Google Scholar 

  47. P. Rosenthal, Equivalents of the invariant subspace problem, Paul Halmos: Celebrating 50 Years of Mathematics, Springer, New York, 1991, 179–188.

    Google Scholar 

  48. A. L. Shields, Weighted shifts operators and analytic function theory, Topics in Operator Theory (Mathematical Surveys No. 13, Amer. Math. Soc., Providence, 2nd pr. 1979), 49–128b.

    Google Scholar 

  49. L. Suciu, Some invariant subspaces for A-contractions and applications, Extracta Math., 21 (2006), 221–247.

    MathSciNet  MATH  Google Scholar 

  50. L. Suciu, Maximum A-isometric part of an A-contraction and applications, Israel J. Math., 174 (2009), 419–443.

    MathSciNet  MATH  Google Scholar 

  51. L. Suciu and N. Suciu, Asymptotic behaviours and generalized Toeplitz operators, J. Math. Anal. Appl., 349 (2009), 280–290.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Catholic University, 22453-900, Rio de Janeiro, RJ, Brazil

    Carlos S. Kubrusly

Authors
  1. Carlos S. Kubrusly
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Carlos S. Kubrusly.

Additional information

Communicated by L. Kérchy

A preliminary draft of this paper was presented at SOTA2 Conference held in Rio in 2001.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kubrusly, C.S. Contractions T for which A is a projection. ActaSci.Math. 80, 603–624 (2014). https://doi.org/10.14232/actasm-013-255-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.14232/actasm-013-255-6

Key words and phrases

AMS Subject Classifications

Search

Navigation