Ranks of Cross-Commutators and Unitary Module Maps



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)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Chen, X., Guo, K.: Analytic Hilbert Modules. Chapman & Hall/CRC, Boca Raton (2003)CrossRefMATHGoogle Scholar
  3. 3.
    Douglas, R., Foias, C.: Uniqueness of multi-variate canonical models. Acta Sci. Math. (Szeged) 57, 73–81 (1993)MathSciNetMATHGoogle Scholar
  4. 4.
    Hoffman, K.: Banach Spaces of Analytic Functions. Prentice-Hall, New Jersey (1962)MATHGoogle 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)MathSciNetMATHGoogle 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)MathSciNetMATHGoogle 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)MathSciNetMATHGoogle Scholar
  8. 8.
    Mandrekar, V.: The validity of Beurling theorems in polydiscs. Proc. Am. Math. Soc. 103, 145–148 (1988)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Nakazi, T.: Invariant subspaces in the bidisc and commutators. J. Aust. Math. Soc. (Ser. A) 56, 232–242 (1994)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Rudin, W.: Function Theory in Polydiscs. Benjamin, New York (1969)MATHGoogle Scholar
  11. 11.
    Yang, R.: The Berger–Shaw theorem in the Hardy module over the bidisk. J. Oper. Theory 42, 379–404 (1999)MathSciNetMATHGoogle Scholar
  12. 12.
    Yang, R.: Operator theory in the Hardy space over the bidisk (III). J. Funct. Anal. 186, 521–545 (2001)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Yang, R.: Hilbert-Schmidt submodules and issues of unitary equivalence. J. Oper. Theory 52, 169–184 (2005)MathSciNetMATHGoogle Scholar
  14. 14.
    Yang, R.: The core operator and congruent submodules. J. Funct. Anal. 228, 469–489 (2005)MathSciNetCrossRefMATHGoogle 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

Personalised recommendations