On the second parameter of an $(m, p)$-isometry

A bounded linear operator $T$ on a Banach space $X$ is called an $(m, p)$-isometry if it satisfies the equation \sum_{k=0}^{m}(-1)^{k} {m \choose k}\|T^{k}x\|^{p} = 0$, for all$x \in X$. In this paper we study the structure which underlies the second parameter of$(m, p)$-isometric operators. We concentrate on determining when an$(m, p)$-isometry is a$(\mu, q)$-isometry for some pair ($\mu, q)$. We also extend the definition of$(m, p)$-isometry, to include$p=\infty$and study basic properties of these$(m, \infty)$-isometries.