On $(m,\infty )$-isometries: Examples

Abstract

An operator T on a Banach space X is said to be an $(m,\infty )$-isometry, if

$$\begin{aligned} \max _{\begin{array}{c} 0 \le k \le m\\ k \;\; \mathrm{even}\\ \end{array}} \Vert T^kx\Vert =\max _{\begin{array}{c} 0 \le k \le m\\ k \;\; \mathrm{odd} \end{array}} \Vert T^kx\Vert , \end{aligned}$$

for all $x\in X$. In this paper, we study unilateral weighted shift operators which are $(m,\infty )$-isometries for some integers m. In particular, we show that any power of an $(m,\infty )$-isometry is not necessarily an $(m,\infty )$-isometry. We also study strict $(3,\infty )$-isometries on ${{\mathbb {R}}}^2$ and give an example of a strict $(2n-1, \infty )$-isometry on ${\mathbb {C}}^2$, for any odd integer n.