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Doubly commuting invariant subspaces for representations of product systems of \(C^*\)-correspondences

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Abstract

We obtain a Shimorin Wold-type decomposition for a doubly commuting covariant representation of a product system of \(C^*\)-correspondences over \({\mathbb {N}}_0^k\). This result gives Shimorin-type decompositions of recent Wold-type decompositions by Jeu and Pinto (Adv Math 368:107–149, 2020) for the q-doubly commuting isometries and by Popescu (J Funct Anal 279:108798, 2020) for Doubly \(\Lambda \)-commuting row isometries. Application to the wandering subspaces of the induced representations is explored, and a version of the Beurling–Lax-type characterization is obtained to study doubly commuting invariant subspaces.

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Acknowledgements

We thank anonymous referee for careful reading of the manuscript and providing many insightful comments and suggestions which helped to improve the exposition. Shankar Veerabathiran is supported by CSIR Fellowship (File No: 09/115(0782)/2017-EMR-I). He is grateful to The LNM Institute of Information Technology for providing research facility and warm hospitality during a visit in December 2019.

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Authors and Affiliations

  1. The LNM Institute of Information Technology, Rupa ki Nangal, Post-Sumel, Via-Jamdoli, Jaipur, Rajasthan, 302031, India

    Harsh Trivedi

  2. Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, 600005, India

    Shankar Veerabathiran

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  1. Harsh Trivedi
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  2. Shankar Veerabathiran
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Correspondence to Harsh Trivedi.

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Communicated by Baruch Solel.

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Trivedi, H., Veerabathiran, S. Doubly commuting invariant subspaces for representations of product systems of \(C^*\)-correspondences. Ann. Funct. Anal. 12, 47 (2021). https://doi.org/10.1007/s43034-021-00136-7

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