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Irreducible Tuples Without the Boundary Property

Published online by Cambridge University Press:  20 November 2018

Sameer Chavan
Sameer Chavan*
Affiliation:
Indian Institute of Technology Kanpur, Kanpur- 208016, India. e-mail: chavan@iitk.ac.in
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Abstract

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We examine spectral behavior of irreducible tuples that do not admit the boundary property. In particular, we prove under some mild assumption that the spectral radius of such an $m$-tuple $\left( {{T}_{1}},\,.\,.\,.\,,\,{{T}_{m}} \right)$must be the operator norm of $T_{1}^{*}\,{{T}_{1}}\,+\,.\,.\,.\,+\,T_{m}^{*}{{T}_{m}}$. We use this simple observation to ensure the boundary property for an irreducible, essentially normal, joint q-isometry, provided it is not a joint isometry. We further exhibit a family of reproducing Hilbert $\mathbb{C}\left[ {{z}_{1}},\,.\,.\,.\,,{{z}_{m}} \right]$-modules (of which the Drury–Arveson Hilbert module is a prototype) with the property that any two nested unitarily equivalent submodules are indeed equal.

Keywords

47A1346E22boundary representationssubnormaljoint p-isometry
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 1 , 01 March 2015 , pp. 9 - 18
Copyright
Copyright © Canadian Mathematical Society 2015

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