  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 $(T_1,T_2)$.
