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 (μ, q)-isometry for some pair (μ, q). We also extend the definition of (m, p)-isometry, to include p = ∞ and study basic properties of these (m, ∞)-isometries.