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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 \(\theta \).

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Acknowledgements

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

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Correspondence to Kei Ji Izuchi.

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Communicated by Raymond Mortini.

The Kei Ji Izuchi is partially supported by Grant-in-Aid for Scientific Research, Japan Society for the Promotion of Science (No. 15K04895).

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Izuchi, K.J., Izuchi, K.H. & Izuchi, Y. Ranks of Cross-Commutators and Unitary Module Maps. Comput. Methods Funct. Theory 18, 545–566 (2018). https://doi.org/10.1007/s40315-018-0239-1

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