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# Irreducible Tuples Without the Boundary Property

Published:2014-11-03
Printed: Mar 2015
• Sameer Chavan,
Indian Institute of Technology Kanpur, Kanpur-208016, India
 Format: LaTeX MathJax PDF

## Abstract

We examine spectral behavior of irreducible tuples which do not admit boundary property. In particular, we prove under some mild assumption that the spectral radius of such an $m$-tuple $(T_1, \dots, T_m)$ must be the operator norm of $T^*_1T_1 + \cdots + T^*_mT_m$. We use this simple observation to ensure 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 $\mathbb{C}[z_1, \dots, z_m]$-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: boundary representations, subnormal, joint p-isometry
 MSC Classifications: 47A13 - Several-variable operator theory (spectral, Fredholm, etc.) 46E22 - Hilbert spaces with reproducing kernels (= [proper] functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) [See also 47B32]

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