A note on operator tuples which are $(m,p)$-isometric as well as $(\mu,\infty)$-isometric

TL;DR

This paper investigates the properties of operator tuples that are both $(m,p)$-isometric and $(,)$-isometric, revealing that their powers form a $(1,p)$-isometry and exploring additional operator characteristics.

Contribution

It establishes a connection between $(m,p)$-isometric and $(,)$-isometric tuples, showing their powers are $(1,p)$-isometric, with new insights for commuting pairs.

Findings

$(T_1^m,...,T_d^m)$ are $(1,p)$-isometries

Additional properties of the operators are derived

Stronger results are obtained for commuting pairs

Abstract

We show that if a tuple of commuting, bounded linear operators $(T_1,...,T_d) \in B(X)^d$ is both an $(m,p)$-isometry and a $(\mu,\infty)$-isometry, then the tuple $(T_1^m,...,T_d^m)$ is a $(1,p)$-isometry. We further prove some additional properties of the operators $T_1,...,T_d$ and show a stronger result in the case of a commuting pair $(T_1,T_2)$.