 

Jaydeb Sarkar, Harsh Trivedi, and Shankar Veerabathiran
Covariant representations of subproduct systems: Invariant subspaces and curvature view print


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 


Abstract
Let X=(X(n))_{n ∈ Z+} be a standard subproduct system
of C*correspondences over a C*algebra M. Let
T=(T_{n})_{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:
F_{X}⊗_{π}D→ H such that
Π (S_{n}(ζ)⊗ I_{D})=T_{n}(ζ)Π
(ζ∈ X(n), n ∈ Z_{+}),
and S = ran Π, or equivalently, P_{S}=ΠΠ*. This result leads us to a list of consequences
including BeurlingLaxHalmos 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.


Acknowledgements
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 DSTInspire fellowship.


Author information
Jaydeb Sarkar:
Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore center, 8th Mile, Mysore Road, Bangalore, 560059, India
jaydeb@gmail.com
Harsh Trivedi:
Silver Oak College of Engineering and Technology, Near Bhagwat Vidyapith, Ahmedabad380061, India.
trivediharsh26@gmail.com
Shankar Veerabathiran:
Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai (Madras) 600005, India
shankarunom@gmail.com

