Computational Methods and Function Theory

, Volume 18, Issue 3, pp 545–566 | Cite as

Ranks of Cross-Commutators and Unitary Module Maps

  • Kei Ji Izuchi
  • Kou Hei Izuchi
  • Yuko Izuchi


Let M be an invariant subspace of \(H^2\) over the bi-disk and \(N=H^2\ominus M\). Let \(S_{z,N},S_{w,N}\) be the compression of the multiplication operators \(T_z,T_w\) on \(H^2\) onto N. For a two-variable inner function \(\theta \), let \(M_\theta = \theta M\) and \(N_\theta =H^2\ominus M_\theta \). We shall study the relationship of the ranks of the cross-commutators \([S_{z,N},S^*_{w,N}]\) and \([S_{z,N_\theta },S^*_{w,N_\theta }]\). We also characterize M such that rank \([S_{z,N},S^*_{w,N}]\) \(\not =\) rank \([S_{z,N_\theta },S^*_{w,N_\theta }]\) for any non-constant inner function \(\theta \).


Hardy space over the bi-disk Invariant subspace Backward shift invariant subspace Unitary module map Rank of cross-commutator 

Mathematics Subject Classification

Primary 47A15 32A35 Secondary 47B35 



The authors would like to thank the referees for their many comments and suggestions.


  1. 1.
    Agrawal, O., Clark, D., Douglas, R.: Invariant subspaces in the polydisk. Pac. J. Math. 121, 1–11 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chen, X., Guo, K.: Analytic Hilbert Modules. Chapman & Hall/CRC, Boca Raton (2003)CrossRefzbMATHGoogle Scholar
  3. 3.
    Douglas, R., Foias, C.: Uniqueness of multi-variate canonical models. Acta Sci. Math. (Szeged) 57, 73–81 (1993)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Hoffman, K.: Banach Spaces of Analytic Functions. Prentice-Hall, New Jersey (1962)zbMATHGoogle Scholar
  5. 5.
    Izuchi, K.J., Izuchi, K.H.: Commutativity in two-variable Jordan blocks on the Hardy space. Acta Sci. Math. (Szeged) 78, 129–136 (2012)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Izuchi, K.J., Nakazi, T., Seto, M.: Backward shift invariant subspaces in the bidisc II. J. Oper. Theory 51, 361–376 (2004)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Izuchi, K.J., Nakazi, T., Seto, M.: Backward shift invariant subspaces in the bidisc III. Acta Sci. Math. (Szeged) 70, 727–749 (2004)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Mandrekar, V.: The validity of Beurling theorems in polydiscs. Proc. Am. Math. Soc. 103, 145–148 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Nakazi, T.: Invariant subspaces in the bidisc and commutators. J. Aust. Math. Soc. (Ser. A) 56, 232–242 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Rudin, W.: Function Theory in Polydiscs. Benjamin, New York (1969)zbMATHGoogle Scholar
  11. 11.
    Yang, R.: The Berger–Shaw theorem in the Hardy module over the bidisk. J. Oper. Theory 42, 379–404 (1999)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Yang, R.: Operator theory in the Hardy space over the bidisk (III). J. Funct. Anal. 186, 521–545 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Yang, R.: Hilbert-Schmidt submodules and issues of unitary equivalence. J. Oper. Theory 52, 169–184 (2005)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Yang, R.: The core operator and congruent submodules. J. Funct. Anal. 228, 469–489 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsNiigata UniversityNiigataJapan
  2. 2.Department of Mathematics, Faculty of EducationYamaguchi UniversityYamaguchiJapan
  3. 3.YamaguchiJapan

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