Skip to main content
SpringerLink
Log in

Operators Cauchy Dual to 2-Hyperexpansive Operators: The Multivariable Case

  • Published:
Integral Equations and Operator Theory Aims and scope Submit manuscript

Abstract

As a natural outgrowth of the work done in Chavan (Proc Edin Math Soc 50:637–652, 2007; Studia Math 203:129–162, 2011), we introduce an abstract framework to study generating m-tuples, and use it to analyze hypercontractivity and hyperexpansivity in several variables. These two notions encompass (joint) hyponormality and subnormality, as well as toral and spherical isometric-ness; for instance, the Drury–Arveson 2-shift is a spherical complete hyperexpansion. Our approach produces a unified theory that simultaneously covers toral and spherical hypercontractions (and hyperexpansions). As a byproduct, we arrive at a dilation theory for completely hypercontractive and completely hyperexpansive generating tuples. We can then analyze in detail the Cauchy duals of toral and spherical 2-hyperexpansive tuples. We also discuss various applications to the theory of hypercontractive and hyperexpansive tuples.

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

Access this article

Log in via an institution

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. Agler, J.: An abstract approach to model theory. In: Surveys of Some Recent Results in Operator Theory, vol. II. Pitman Res. Notes Math. Ser., vol. 192, pp. 1–23. Longman Sci. Tech., Harlow (1988)

  2. Aleman, A.: The multiplication operators on Hilbert spaces of analytic functions. Habilitationsschrift

  3. Aronszajn N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arveson W.: Subalgebras of C*-algebras. III. Multivariable operator theory. Acta Math. 181, 159–228 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Arveson W.: The curvature invariant of a Hilbert module over \({C[z_1,\dots,z_d]}\) . J. Reine Angew. Math. 522, 173–236 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Arveson W.: The Dirac operator of a commuting d-tuple. J. Funct. Anal. 189, 53–79 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Arveson W.: Dilation theory yesterday and today. A glimpse at Hilbert space operators. Oper. Theory Adv. Appl. 207, 99–123 (2010)

    MathSciNet  Google Scholar 

  8. Athavale A.: Holomorphic kernels and commuting operators. Trans. Am. Math. Soc. 304, 101–110 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  9. Athavale A.: Model theory on the unit ball in \({\mathbb{C}^{n}}\) . J. Oper. Theory 27, 347–358 (1992)

    MathSciNet  MATH  Google Scholar 

  10. Athavale A.: On a minimal normal dilation problem. Proc. Am. Math. Soc. 121, 843–850 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Athavale A.: On completely hyperexpansive operators. Proc. Am. Math. Soc. 124, 3745–3752 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  12. Athavale A.: The complete hyperexpansivity analog of the Embry conditions. Studia Math. 154, 233–242 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Athavale A., Pedersen S.: Moment problems and subnormality. J. Math. Anal. Appl. 146, 434–441 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  14. Athavale A., Sholapurkar V.: Completely hyperexpansive operator tuples. Positivity 3, 245–257 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bayart F., Matheron É.: Dynamics of Linear Operators. Cambridge University Press, Cambridge (2009)

    Book  MATH  Google Scholar 

  16. Berg C., Christensen J.P.R., Ressel P.: Harmonic Analysis on Semigroups. Springer, Berlin (1984)

    Book  MATH  Google Scholar 

  17. Bunce J.: The joint spectrum of commuting nonnormal operators. Proc. Am. Math. Soc. 29, 499–505 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chavan S.: On operators Cauchy dual to 2-hyperexpansive operators. Proc. Edinb. Math. Soc. 50, 637–652 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chavan S.: On operators close to isometries. Studia Math. 186, 275–293 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chavan S.: Co-analytic, right-invertible operators are supercyclic. Colloq. Math. 119, 137–142 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chavan S.: On operators Cauchy dual to 2-hyperexpansive operators: the unbounded case. Studia Math. 203, 129–162 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Chō M., Żelazko W.: On geometric spectral radius of commuting n-tuples of operators. Hokkaido Math. J. 21, 251–258 (1992)

    MathSciNet  MATH  Google Scholar 

  23. Conway J.: A Course in Functional Analysis. Springer, New York (1997)

    Google Scholar 

  24. Conway, J.: The Theory of Subnormal Operators. Math. Surveys Monographs, vol. 36. Amer. Math. Soc., Providence (1991)

  25. Curto, R.: Applications of several complex variables to multiparameter spectral theory. In: Surveys of Some Recent Results in Operator Theory, vol. II. Pitman Res. Notes Math. Ser., vol. 192, pp. 25–90. Longman Sci. Tech., Harlow (1988)

  26. Curto R.: Joint hyponormality: a bridge between hyponormality and subnormality. Proc. Symposia Pure Math. 51, 69–91 (1990)

    MathSciNet  Google Scholar 

  27. Curto, R.: Spectral theory of elementary operators. In: Elementary Operators and Applications (Blaubeuren, 1991), pp. 3–52. World Scientific, River Edge (1992)

  28. Curto R., Vasilescu F.: Standard operator models in the polydisc-I. Indiana Univ. Math. J. 42, 979–989 (1993)

    Article  MathSciNet  Google Scholar 

  29. Curto R., Vasilescu F.: Standard operator models in the polydisc-II. Indiana Univ. Math. J. 44, 727–746 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  30. Curto R., Yan K.: The spectral picture of Reinhardt measures. J. Funct. Anal. 131, 279–301 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  31. Drury S.: A generalization of von Neumann’s inequality to the complex ball. Proc. Am. Math. Soc. 68, 300–304 (1978)

    MathSciNet  MATH  Google Scholar 

  32. Fialkow L.A.: Essential spectra of elementary operators. Trans. Am. Math. Soc. 267, 157–174 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  33. Fialkow L.A.: Spectral properties of elementary operators. Acta Sci. Math. (Szeged) 46, 269–282 (1983)

    MathSciNet  MATH  Google Scholar 

  34. Fialkow L.A.: Spectral properties of elementary operators, II. Trans. Am. Math. Soc. 290, 415–429 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  35. Fialkow, L.A.: Structural properties of elementary operators. In: Elementary Operators and Applications (Blaubeuren, 1991), pp. 55–113. World Scientific, River Edge (1992)

  36. Gleason J., Richter S.: m-Isometric commuting tuples of operators on a Hilbert space. Integr. Equ. Oper. Theory 56, 181–196 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  37. Guo K., Hu J., Xu X.: Toeplitz algebras, subnormal tuples and rigidity on reproducing \({\mathbf{C}[z_1,\dots,z_d]}\) -modules. J. Funct. Anal. 210, 214–247 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  38. Itô T.: On the commutative family of subnormal operators. J. Fac. Sci. Hokkaido Univ. Ser. I 14, 1–15 (1958)

    MathSciNet  MATH  Google Scholar 

  39. Jabloński Z.: Complete hyperexpansivity, subnormality and inverted boundedness conditions. Integr. Equ. Oper. Theory 44, 316–336 (2002)

    Article  MATH  Google Scholar 

  40. Jabloński Z., Stochel J.: Unbounded 2-hyperexpansive operators. Proc. Edinb. Math. Soc. 44, 613–629 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  41. Kubrusly C.: An Introduction to Models and Decompositions in Operator Theory. Birkhäuser, Boston (1997)

    Book  MATH  Google Scholar 

  42. Müller V., Soltysiak A.: Spectral radius formula for commuting Hilbert space operators. Studia Math. 103, 329–333 (1992)

    MathSciNet  MATH  Google Scholar 

  43. Müller V., Vasilescu F.H.: Standard models for some commuting multioperators. Proc. Am. Math. Soc. 117, 979–989 (1993)

    Article  MATH  Google Scholar 

  44. Olofsson A.: Wandering subspace theorems. Integr. Equ. Oper. Theory 51, 395–409 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  45. Magajna B.: A transitivity theorem for algebras of elementary operators. Proc. Am. Math. Soc. 118, 119–127 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  46. Magajna B.: Interpolation by elementary operators. Studia Math. 105, 77–92 (1993)

    MathSciNet  MATH  Google Scholar 

  47. Magajna B.: The norm of a symmetric elementary operator. Proc. Am. Math. Soc. 132, 1747–1754 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  48. Magajna, B.: Approximation of maps on C*-algebras by completely contractive elementary operators. In: Elementary Operators and Their Applications. Oper. Theory Adv. Appl., vol. 212, pp. 25–39 (2011)

  49. Mathieu M.: Elementary operators on prime C*-algebras, II. Glasgow Math. J. 30, 275–284 (1988)

    Article  MathSciNet  Google Scholar 

  50. Mathieu M.: Elementary operators on prime C*-algebras, I. Math. Ann. 284, 223–244 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  51. Mathieu, M.: How to use primeness to describe properties of elementary operators. In: Operator Theory: Operator Algebras and Applications, Part 2 (Durham, NH, 1988). Proc. Sympos. Pure Math., vol. 51, Part 2, pp. 195–199. Amer. Math. Soc., Providence (1990)

  52. Mathieu, M.: Operator equations with elementary operators. In: Multivariable Operator Theory (Seattle, WA, 1993). Contemporary Math., vol. 185, pp. 259–272 (1995)

  53. Mathieu, M.: The norm problem for elementary operators. In: Recent Progress in Functional Analysis (Valencia, 2000). North Holland Math. Stud., vol. 189, pp. 363–368. North-Holland, Amsterdam (2001)

  54. Paulsen V.: Completely Bounded Maps and Operator Algebras. Cambridge University Press, Cambridge (2003)

    Book  Google Scholar 

  55. Richter S.: Invariant subspaces of the Dirichlet shift. J. Reine Angew. Math. 386, 205–220 (1988)

    MathSciNet  MATH  Google Scholar 

  56. Rudin W.: Real and Complex Analysis. McGraw-Hill, New York (1987)

    MATH  Google Scholar 

  57. Saksman E., Tylli, H.-O.: Multiplications and elementary operators in the Banach space setting. In: Methods in Banach Space Theory. London Math. Soc. Lecture Note Ser., vol. 337, pp. 253–292. Cambridge University Press, Cambridge (2006)

  58. Shimorin S.: Wold-type decompositions and wandering subspaces for operators close to isometries. J. Reine Angew. Math. 531, 147–189 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  59. Shimorin S.: Complete Nevanlinna-Pick property of Dirichlet-type spaces. J. Funct. Anal. 191, 276–296 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  60. Sholapurkar V., Athavale A.: Completely and alternatingly hyperexpansive operators. J. Oper. Theory 43, 43–68 (2000)

    MathSciNet  MATH  Google Scholar 

  61. Stochel J.: Characterizations of subnormal operators. Studia Math. 97, 227–238 (1991)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, 208016, India

    Sameer Chavan

  2. Department of Mathematics, The University of Iowa, Iowa City, IA, 52242-1419, USA

    Raúl E. Curto

Authors
  1. Sameer Chavan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Raúl E. Curto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Raúl E. Curto.

Additional information

R.E. Curto’s research was partially supported by NSF Research Grant DMS-0801168.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chavan, S., Curto, R.E. Operators Cauchy Dual to 2-Hyperexpansive Operators: The Multivariable Case. Integr. Equ. Oper. Theory 73, 481–516 (2012). https://doi.org/10.1007/s00020-012-1986-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00020-012-1986-4

Mathematics Subject Classification (2010)

Keywords

Search

Navigation