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A Characterization of Invariant Subspaces for Isometric Representations of Product System over \(\mathbb {N}_0^{k}\)

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Abstract

Using the Wold–von Neumann decomposition for the isometric covariant representations due to Muhly and Solel, we prove an explicit representation of the commutant of a doubly commuting pure isometric representation of the product system over \(\mathbb {N}_0^{k}.\) As an application we study a complete characterization of invariant subspaces for a doubly commuting pure isometric representation of the product system. This provides us a complete set of isomorphic invariants. Finally, we classify a large class of an isometric covariant representations of the product system.

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Acknowledgements

We thank the reviewer and the editor for suggesting changes in the introduction. Dimple Saini is supported by UGC fellowship (File No:16-6(DEC. 2018)/2019(NET/CSIR)). Harsh Trivedi is supported by MATRICS-SERB Research Grant, File No: MTR/2021/000286, by SERB, Department of Science & Technology (DST), Government of India. Shankar Veerabathiran thanks IMSc Chennai for postdoc fellowship. Saini and Trivedi acknowledge the DST-FIST program (Govt. of India) FIST - No. SR/FST/MS-I/2018/24.

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Authors and Affiliations

  1. Centre for Mathematical and Financial Computing, Department of Mathematics, The LNM Institute of Information Technology, Via-Jamdoli, Jaipur, Rajasthan, 302031, India

    Dimple Saini & Harsh Trivedi

  2. The Institute of Mathematical Sciences, A CI of Homi Bhabha National Institute, CIT Campus Taramani, Chennai, Tamilnadu, 600113, India

    Shankar Veerabathiran

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  1. Dimple Saini
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Saini, D., Trivedi, H. & Veerabathiran, S. A Characterization of Invariant Subspaces for Isometric Representations of Product System over \(\mathbb {N}_0^{k}\). Complex Anal. Oper. Theory 18, 75 (2024). https://doi.org/10.1007/s11785-024-01520-6

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