New York Journal of Mathematics
Volume 24 (2018) 211-232


Jaydeb Sarkar, Harsh Trivedi, and Shankar Veerabathiran

Covariant representations of subproduct systems: Invariant subspaces and curvature

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Published: February 24, 2018
Keywords: Hilbert C*-modules, covariant representations, subproduct systems, tuples of operators, invariant subspaces, wandering subspaces, curvatures
Subject: 46L08, 47A13, 47A15, 47B38, 47L30, 47L55, 47L80

Let X=(X(n))n ∈ Z+ be a standard subproduct system of C*-correspondences over a C*-algebra M. Let T=(Tn)n ∈ Z+ be a pure completely contractive, covariant representation of X on a Hilbert space H. If S is a closed subspace of H, then S is invariant for T if and only if there exist a Hilbert space D, a representation π of M on D, and a partial isometry Π: FXπDH such that
Π (Sn(ζ)⊗ ID)=Tn(ζ)Π    (ζ∈ X(n), n ∈ Z+),
and S = ran Π, or equivalently, PS=ΠΠ*. This result leads us to a list of consequences including Beurling-Lax-Halmos type theorem and other general observations on wandering subspaces. We extend the notion of curvature for completely contractive, covariant representations and analyze it in terms of the above results.


The research of Sarkar was supported in part by (1) National Board of Higher Mathematics (NBHM), India, grant NBHM/R.P.64/2014, and (2) Mathematical Research Impact Centric Support (MATRICS) grant, File No : MTR/2017/000522, by the Science and Engineering Research Board (SERB), Department of Science & Technology (DST), Government of India. Trivedi thanks Indian Statistical Institute Bangalore for the visiting scientist fellowship. Veerabathiran was supported by DST-Inspire fellowship.

Author information

Jaydeb Sarkar:
Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore center, 8th Mile, Mysore Road, Bangalore, 560059, India

Harsh Trivedi:
Silver Oak College of Engineering and Technology, Near Bhagwat Vidyapith, Ahmedabad-380061, India.

Shankar Veerabathiran:
Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai (Madras) 600005, India