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

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 \((T_1,T_2)\).