A bounded linear operator T on a Banach space X is called an (m,p)-isometry for a positive integer m and a real number p ≥ 1 if, for any vector x ∈ X, ${\sum }_{k=0}^m{(-1)}^k\left(\genfrac{}{}{0pt}{}{m}{k}\right)\left|\right|T^{k}x{\left|\right|}^p=0$. We prove that any power of an (m,p)-isometry is also an (m,p)-isometry. In general the converse is not true. However, we prove that if $T^r$ and $T^{r+1}$ are (m,p)-isometries for a positive integer r, then T is an (m,p)-isometry. More precisely, if $T^r$ is an (m,p)-isometry and $T^s$ is an (l,p)-isometry, then $T^t$ is an (h,p)-isometry, where t = gcd(r,s) and h = min(m,l)....