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1. CMB Online first
Irreducible Tuples without Boundary Property 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 Categories:47A13, 46E22 |
2. CMB 2011 (vol 54 pp. 680)
$2$-Local Isometries on Spaces of Lipschitz Functions Let $(X,d)$ be a metric space, and let $\mathop{\textrm{Lip}}(X)$ denote the Banach
space of all scalar-valued bounded Lipschitz functions $f$ on $X$
endowed with one of the natural norms
$
\| f\| =\max \{\| f\| _\infty ,L(f)\}$ or $\|f\| =\|
f\| _\infty +L(f),
$
where $L(f)$ is the
Lipschitz constant of $f.$ It is said that the isometry
group of $\mathop{\textrm{Lip}}(X)$ is canonical if every
surjective linear isometry of
$\mathop{\textrm{Lip}}(X) $ is induced by a surjective isometry of $X$.
In this paper
we prove that if $X$ is bounded separable and the isometry group of
$\mathop{\textrm{Lip}}(X)$ is canonical, then every $2$-local isometry
of $\mathop{\textrm{Lip}}(X)$ is
a surjective linear isometry. Furthermore, we give a complete
description of all $2$-local isometries of $\mathop{\textrm{Lip}}(X)$ when $X$ is
bounded.
Keywords:isometry, local isometry, Lipschitz function Categories:46B04, 46J10, 46E15 |
3. CMB 2010 (vol 53 pp. 398)
Projections in the Convex Hull of Surjective Isometries We characterize those linear projections represented as a convex combination of two surjective isometries on standard Banach spaces of continuous functions with values in a strictly convex Banach space.
Keywords:isometry, convex combination of isometries, generalized bi-circular projections Categories:47A65, 47B15, 47B37 |