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

TL;DR

This paper investigates the structure of $(m, p)$-isometric operators on Banach spaces, focusing on when such operators can be characterized as $(d, q)$-isometries and extending the concept to include the case $p=f$.

Contribution

It provides a detailed analysis of the second parameter in $(m, p)$-isometries, including conditions for equivalence to other isometries and an extension to the case $p=f$.

Findings

Characterization of when $(m, p)$-isometries are also $(d, q)$-isometries.

Extension of the definition to include $p=f$ and exploration of properties of these operators.

Abstract

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.