Ranks of Cross-Commutators and Unitary Module Maps

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Abstract

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 \(\theta \).

Keywords

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 

Notes

Acknowledgements

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

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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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