Irreducible Tuples Without the Boundary Property

Chavan, Sameer

Canadian mathematical bulletin, Tome 58 (2015) no. 1, pp. 9-18

Résumé

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.

Détail

DOI : 10.4153/CMB-2014-051-0

Mots-clés : 47A13, 46E22, boundary representations, subnormal, joint p-isometry