On the Second Parameter of an (<Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">p</Emphasis>)-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.