Opuscula Math. 39, no. 4 (2019), 543-555

Opuscula Mathematica

On unitary equivalence of bilateral operator valued weighted shifts

Jakub Kośmider

Abstract. We establish a characterization of unitary equivalence of two bilateral operator valued weighted shifts with quasi-invertible weights by an operator of diagonal form. We provide an example of unitary equivalence between shifts with weights defined on \(\mathbb{C}^2\) which cannot be given by any unitary operator of diagonal form. The paper also contains some remarks regarding unitary operators that can give unitary equivalence of bilateral operator valued weighted shifts.

Keywords: unitary equivalence, bilateral weighted shift, quasi-invertible weights, partial isometry.

Mathematics Subject Classification: 47B37, 47A62.

Full text (pdf)

  1. A. Anand, S. Chavan, Z.J. Jabłoński, J. Stochel, Complete systems of unitary invariants for some classes of 2-isometries, Banach J. Math. Anal. 13 (2019) 2, 359-385.
  2. A. Athavale, On completely hyperexpansive operators, Proc. Amer. Math. Soc. 124 (1996), 3745-3752.
  3. A. Bourhim, C.E. Chidume,The single-valued extension property for bilateral operator weighted shifts, Proc. Amer. Math. Soc., 133 (2004), 485-491.
  4. J.B. Conway, A Course in Functional Analysis, Springer-Verlag, New York, 1990.
  5. G.P. Gehér, Bilateral weighted shift operators similar to normal operators, Oper. Matrices 10 (2016) 2, 419-423.
  6. J. Guyker, On reducing subspaces of normally weighted bilateral shifts, Houston J. Math. 11 (1985) 4, 515-521.
  7. R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, 1990.
  8. N. Ivanovski, Similiarity and quasisimiliarity of bilateral operator valued weighted shifts, Mat. Bilten 17 (1993), 33-37.
  9. Z.J. Jabłoński, Hyperexpansive operator valued unilateral weighted shifts, Glasg. Math. J. 46 (2004), 405-416.
  10. Z.J. Jabłoński, I.B. Jung, J. Stochel, Weighted Shifts on Directed Trees, Mem. Amer. Math. Soc. 216 (2012) 1017.
  11. A. Lambert, Unitary equivalence and reducibility of invertibly weighted shifts, Bull. Aust. Math. Soc. 5 (1971), 157-173.
  12. J.X. Li, Y.Q. Ji, S.L. Sun, The essential spectrum and Banach reducibility of operator weighted shifts, Acta Math. Sin., English Series, 17 (2001) 3, 413-424.
  13. M. Orovčanec, Unitary equivalence of unilateral operator valued weighted shifts with quasi-invertible weights, Mat. Bilten 17 (1993), 45-50.
  14. P. Pietrzycki, The single equality \(A^{*n}A^n=(A^*A)^n\) does not imply the quasinormality of weighted shifts on rootless directed trees, J. Math. Anal. Appl. 435 (2016), 338-348.
  15. V.S. Pilidi, On unitary equivalence of multiple weighted shift operators, Teor. Funkts. Funkts. Anal. Prilozh. 34 (1980), 96-103 [in Russian].
  16. A. Shields, Weighted shift operators, analytic function theory, Topics in Operator Theory, Math. Surveys 13, Amer. Math. Soc., Providence, R. I. (1974), 49-128.
  • Communicated by P.A. Cojuhari.
  • Received: 2019-03-04.
  • Revised: 2019-03-19.
  • Accepted: 2019-03-20.
  • Published online: 2019-05-23.
Opuscula Mathematica - cover

Cite this article as:
Jakub Kośmider, On unitary equivalence of bilateral operator valued weighted shifts, Opuscula Math. 39, no. 4 (2019), 543-555, https://doi.org/10.7494/OpMath.2019.39.4.543

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.