Irreducible Tuples Without the Boundary Property. Issue 1 (1st March 2015) - Chavan, Sameer
Abstract: 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.