Skip to main content
SpringerLink
Log in

Aluthge Transforms of 2-Variable Weighted Shifts

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

Abstract

We introduce two natural notions of multivariable Aluthge transforms (toral and spherical), and study their basic properties. In the case of 2-variable weighted shifts, we first prove that the toral Aluthge transform does not preserve (joint) hyponormality, in sharp contrast with the 1-variable case. Second, we identify a large class of 2-variable weighted shifts for which hyponormality is preserved under both transforms. Third, we consider whether these Aluthge transforms are norm-continuous. Fourth, we study how the Taylor and Taylor essential spectra of 2-variable weighted shifts behave under the toral and spherical Aluthge transforms; as a special case, we consider the Aluthge transforms of the Drury–Arveson 2-shift. Finally, we briefly discuss the class of spherically quasinormal 2-variable weighted shifts, which are the fixed points for the spherical Aluthge transform.

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. Aluthge, A.: On \(p\)-hyponormal operators for \(0<p<1\). Integral Equ. Oper. Theory 13, 307–315 (1990)

    Google Scholar 

  2. Ando, T.: Aluthge transforms and the convex hull of the spectrum of a Hilbert space operator. In: Recent Advances in Operator Theory and its Applications. Oper. Theory Adv. Appl., vol. 160, pp. 21–39 (2005)

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

    Google Scholar 

  4. Athavale, A.: On joint hyponormality of operators. Proc. Am. Math. Soc. 103, 417–423 (1988)

    Article  MathSciNet  Google Scholar 

  5. Athavale, A.: On the intertwining of joint isometries. J. Oper. Theory 23, 339–350 (1990)

    MathSciNet  MATH  Google Scholar 

  6. Athavale, A., Poddar, S.: On the reflexivity of certain operator tuples. Acta Math. Sci. (Szeged) 81, 285–291 (2015)

    Article  MathSciNet  Google Scholar 

  7. Attele, K.R.M., Lubin, A.R.: Commutant lifting for jointly isometric operators—a geometrical approach. J. Funct. Anal. 140, 300–311 (1996)

    Article  MathSciNet  Google Scholar 

  8. Chavan, S., Sholapurkar, V.: Rigidity theorems for spherical hyperexpansions. Complex Anal. Oper. Theory 7, 1545–1568 (2013)

    Article  MathSciNet  Google Scholar 

  9. Cho, M., Jung, I.B., Lee, W.Y.: On Aluthge transforms of \(p\)-hyponormal operators. Integral Equ. Oper. Theory 53, 321–329 (2005)

    Google Scholar 

  10. Conway, J.: The Theory of Subnormal Operators, Mathematical Surveys and Monographs, vol. 36. American Mathematical Society, Providence (1991)

    Book  Google Scholar 

  11. Curto, R.: On the connectedness of invertible n-tuple. Indiana Univ. Math. J. 29, 393–406 (1980)

    Article  MathSciNet  Google Scholar 

  12. Curto, R.: Applications of several complex variables to multi-parameter spectral theory. In: Conway, J.B., Morrel, B.B. (eds.) Surveys of Recent Results in Operator Theory, vol. II, pp. 25–90. Longman Publishing, London (1988)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Curto, R.: Quadratically hyponormal weighted shifts. Integral Equ. Oper. Theory 13, 49–66 (1990)

    Article  MathSciNet  Google Scholar 

  15. Curto, R.: Spectral theory of elementary operators. In: Mathieu, M. (ed.) Elementary Operators and Applications, pp. 3–52. World Sci. Publishing, River Edge (1992)

    Google Scholar 

  16. Curto, R.E., Fialkow, L.A.: The spectral picture of \((L_A, R_B)\). J. Funct. Anal. 71, 371–392 (1987)

    Google Scholar 

  17. Curto, R., Lee, S.H., Yoon, J.: \(k\)-hyponormality of multivariable weighted shifts. J. Funct. Anal. 229, 462–480 (2005)

    Google Scholar 

  18. Curto, R., Lee, S.H., Yoon, J.: Hyponormality and subnormality for powers of commuting pairs of subnormal operators. J. Funct. Anal. 245, 390–412 (2007)

    Article  MathSciNet  Google Scholar 

  19. Curto, R., Lee, S.H., Yoon, J.: Reconstruction of the Berger measure when the core is of tensor form. Actas del XVI Coloquio Latinoamericano de Álgebra, Bibl. Rev. Mat. Iberoamericana, pp. 317–331 (2007)

  20. Curto, R., Lee, S.H., Yoon, J.: Which 2-hyponormal 2-variable weighted shifts are subnormal? Linear Algebra Appl. 429, 2227–2238 (2008)

    Article  MathSciNet  Google Scholar 

  21. Curto, R., Lee, S.H., Yoon, J.: Subnormality of 2-variable weighted shifts with diagonal core. C. R. Acad. Sci. Paris 351, 203–207 (2013)

    Article  MathSciNet  Google Scholar 

  22. Curto, R., Muhly, P., Xia, J.: Hyponormal pairs of commuting operators. Oper. Theory Adv. Appl. 35, 1–22 (1988)

    MathSciNet  MATH  Google Scholar 

  23. Curto, R., Poon, Y.T., Yoon, J.: Subnormality of Bergman-like weighted shifts. J. Math. Anal. Appl. 308, 334–342 (2005)

    Article  MathSciNet  Google Scholar 

  24. Curto, R., Yoon, J.: Jointly hyponormal pairs of subnormal operators need not be jointly subnormal. Trans. Am. Math. Soc. 358, 5139–5159 (2006)

    Article  MathSciNet  Google Scholar 

  25. Curto, R., Yoon, J.: Disintegration-of-measure techniques for multivariable weighted shifts. Proc. Lond. Math. Soc. 93, 381–402 (2006)

    Article  MathSciNet  Google Scholar 

  26. Curto, R., Yoon, J.: Propagation phenomena for hyponormal 2-variable weighted shifts. J. Oper. Theory 58, 101–130 (2007)

    MathSciNet  MATH  Google Scholar 

  27. Curto, R., Yoon, J.: Spectral picture of 2-variable weighted shifts. C. R. Acad. Sci. Paris 343, 579–584 (2006)

    Article  MathSciNet  Google Scholar 

  28. Curto, R., Yoon, J.: Toral and spherical Aluthge transforms of 2-variable weighted shifts. C. R. Acad. Sci. Paris 354, 1200–1204 (2016)

    Article  MathSciNet  Google Scholar 

  29. Dykema, K., Schultz, H.: Brown measure and iterates of the Aluthge transform for some operators arising from measurable actions. Trans. Am. Math. Soc. 361, 6583–6593 (2009)

    Article  MathSciNet  Google Scholar 

  30. Eschmeier, J., Putinar, M.: Some remarks on spherical isometries. Oper. Theory Adv. Appl. 129, 271–291 (2001)

    MathSciNet  MATH  Google Scholar 

  31. Exner, G.: Subnormality of transformations of Bergman-like weighted shifts. In: Special Session on Recent Advances in Hilbert Space Operators, International Workshop on Operator Theory and Applications (IWOTA), Seoul, Korea, July 31–August 3 (2006)

  32. Exner, G.R.: Aluthge transforms and \(n\)-contractivity of weighted shifts. J. Oper. Theory 61, 419–438 (2009)

    Google Scholar 

  33. Gellar, R., Wallen, L.J.: Subnormal weighted shifts and the Halmos–Bram criterion. Proc. Jpn. Acad. 46, 375–378 (1970)

    Article  MathSciNet  Google Scholar 

  34. Gleason, J.: Quasinormality of Toeplitz tuples with analytic symbols. Houst. J. Math. 32, 293–298 (2006)

    MathSciNet  MATH  Google Scholar 

  35. Gleason, J.: Quasinormality and commuting tuples, preprint (2004)

  36. Jewell, N.P., Lubin, A.R.: Commuting weighted shifts and analytic function theory in several variables. J. Oper. Theory 1, 207–223 (1979)

    MathSciNet  MATH  Google Scholar 

  37. Jung, I.B., Ko, E., Pearcy, C.: Aluthge transform of operators. Integral Equ. Oper. Theory 37, 437–448 (2000)

    Article  MathSciNet  Google Scholar 

  38. Jung, I.B., Ko, E., Pearcy, C.: Spectral pictures of Aluthge transforms of operators. Integral Equ. Oper. Theory 40, 52–60 (2001)

    Article  MathSciNet  Google Scholar 

  39. Kim, J., Yoon, J.: Schur product techniques for the subnormality of commuting 2-variable weighted shifts. Linear Algebra Appl. 453, 174–191 (2014)

    Article  MathSciNet  Google Scholar 

  40. Kim, M.K., Ko, E.: Some connections between an operator and its Aluthge transform. Glasg. Math. J. 47, 167–175 (2005)

    Article  MathSciNet  Google Scholar 

  41. Kimura, F.: Analysis of non-normal operators via Aluthge transformation. Integral Equ. Oper. Theory 50, 375–384 (2004)

    Article  MathSciNet  Google Scholar 

  42. Lee, S.H., Lee, W.Y., Yoon, J.: Subnormality of Aluthge transform of weighted shifts. Integral Equ. Oper. Theory 72, 241–251 (2012)

    Article  MathSciNet  Google Scholar 

  43. Müller, V., Ptak, M.: Spherical isometries are hyporeflexive. Rocky Mount. J. Math. 29, 677–683 (1999)

    Article  MathSciNet  Google Scholar 

  44. Taylor, J.L.: A joint spectrum for several commuting operators. J. Funct. Anal. 6, 172–191 (1970)

    Article  MathSciNet  Google Scholar 

  45. Taylor, J.L.: The analytic functional calculus for several commuting operators. Acta Math. 125, 1–48 (1970)

    Article  MathSciNet  Google Scholar 

  46. Wolfram Research, Inc., Mathematica, Version 9.0, Champaign, IL (2013)

  47. Yamazaki, T.: An expression of spectral radius via Aluthge transformation. Proc. Am. Math. Soc. 130, 1131–1137 (2002)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Preliminary (and substantially abridged) versions of some results in this paper were announced in [28]. Some of the calculations in this paper were obtained using the software tool Mathematica [46].

Author information

Authors and Affiliations

  1. Department of Mathematics, The University of Iowa, Iowa City, IA , 52242, USA

    Raúl E. Curto

  2. School of Mathematical and Statistical Sciences, The University of Texas Rio Grande Valley, Edinburg, TX , 78539, USA

    Jasang Yoon

Authors
  1. Raúl E. Curto
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jasang Yoon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Raúl E. Curto.

Additional information

The first named author was partially supported by NSF Grants DMS-0801168 and DMS-1302666. The second named author was partially supported by the University of Texas System and the Consejo Nacional de Ciencia y Tecnología de Méjico (CONACYT).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Curto, R.E., Yoon, J. Aluthge Transforms of 2-Variable Weighted Shifts. Integr. Equ. Oper. Theory 90, 52 (2018). https://doi.org/10.1007/s00020-018-2475-1

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00020-018-2475-1

Mathematics Subject Classification

Keywords

Search

Navigation