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Rank of a co-doubly commuting submodule is $ 2$


Authors: Arup Chattopadhyay, B. Krishna Das and Jaydeb Sarkar
Journal: Proc. Amer. Math. Soc. 146 (2018), 1181-1187
MSC (2010): Primary 47A13, 47A15, 47A16, 46M05, 46C99, 32A70
DOI: https://doi.org/10.1090/proc/13792
Published electronically: October 23, 2017
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Abstract: We prove that the rank of a non-trivial co-doubly commuting submodule is $ 2$. More precisely, let $ \varphi , \psi \in H^\infty (\mathbb{D})$ be two inner functions. If $ \mathcal {Q}_{\varphi } = H^2(\mathbb{D})/ \varphi H^2(\mathbb{D})$ and $ \mathcal {Q}_{\psi } = H^2(\mathbb{D})/ \psi H^2(\mathbb{D})$, then

$\displaystyle \mbox {rank~}(\mathcal {Q}_{\varphi } \otimes \mathcal {Q}_{\psi })^\perp = 2. $

An immediate consequence is the following: Let $ \mathcal {S}$ be a co-doubly commuting submodule of $ H^2(\mathbb{D}^2)$. Then $ \mbox {rank~} \mathcal {S} = 1$ if and only if $ \mathcal {S} = \Phi H^2(\mathbb{D}^2)$ for some one variable inner function $ \Phi \in H^\infty (\mathbb{D}^2)$. This answers a question posed by R. G. Douglas and R. Yang [Integral Equations Operator Theory 38(2000), pp207-221]

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Additional Information

Arup Chattopadhyay
Affiliation: Department of Mathematics, Indian Institute of Technology Guwahati, Amingaon Post, Guwahati 781039 Assam, India
Email: arupchatt@iitg.ernet.in, 2003arupchattopadhyay@gmail.com

B. Krishna Das
Affiliation: Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
Email: dasb@math.iitb.ac.in, bata436@gmail.com

Jaydeb Sarkar
Affiliation: Statistics and Mathematics Unit, Indian Statistical Institute, 8th Mile, Mysore Road, Bangalore 560059, India
Email: jay@isibang.ac.in, jaydeb@gmail.com

DOI: https://doi.org/10.1090/proc/13792
Keywords: Hardy space over bidisc, rank, joint invariant subspaces, semi-invariant subspaces
Received by editor(s): February 7, 2017
Received by editor(s) in revised form: April 25, 2017
Published electronically: October 23, 2017
Dedicated: Dedicated to the memory of our friend and colleague Sudipta Dutta
Communicated by: Stephan Ramon Garcia
Article copyright: © Copyright 2017 American Mathematical Society

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