Communications of the
Korean Mathematical Society CKMS

Let $p\ge 1$ be a real number. A tuple ${T}=(T_1,\ldots ,T_n)$ of commuting bounded linear operators on a Banach space $X$ is called an $\ell^p$-spherical isometry if $\sum_{i=1} ^n \|T_ix\|^p =\|x\|^p$ for all $x\in X$. The tuple $T$ is called a toral isometry if each $T_i$ is an isometry. By a result of Ansari, Hedayatian, Khani-Robati and Moradi, for every $n\ge1$, there is a supercyclic $\ell^2$-spherical isometric $n$-tuple on ${\mathbb C}^{n}$ but there is no supercyclic $\ell^2$-spherical isometry on an infinite-dimensional Hilbert space. In this article, we investigate the supercyclicity of $\ell^p$-spherical isometries and toral isometries on Banach spaces. Also, we introduce the notion of semi-commutative tuples and we show that the Banach spaces $\ell^p$ ($1\le p<\infty$) support supercyclic $\ell^p$-spherical isometric semi-commutative tuples. As a result, all separable infinite-dimensional complex Hilbert spaces support supercyclic spherical isometric semi-commutative tuples.

Keywords: spherical isometry, toral isometry, supercyclic

MSC numbers: Primary 47A16; Secondary 47A15