Abstract

Let H be a separable complex Hilbert space. A commuting tuple <TEX>$T=(T_1,{\cdots},T_n)$</TEX> of bounded linear operators on H is called a spherical isometry if <TEX>$\sum_{i=1}^{n}T^*_iT_i=I$</TEX>. The tuple T is called a toral isometry if each <TEX>$T_i$</TEX> is an isometry. In this paper, we show that for each <TEX>$n{\geq}1$</TEX> there is a supercyclic n-tuple of spherical isometries on <TEX>$\mathbb{C}^n$</TEX> and there is no spherical or toral isometric tuple of operators on an infinite-dimensional Hilbert space.

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