Abstract. We prove that if $T$ is an $m$-isometry on a Hilbert space and $b(z)$ is an inner function, then $b(T)$ is also an $m$-isometry. This work is motivated by Bermúdez, Mendoza and Martinón [BMM] where it was proved that if $T$ is an $(m,p)$-isometry on a Banach space, then $T^{r}$ is also an $(m,p)$-isometry for any positive integer $r.$ We also prove several functional calculus formulas for a single operator or the product of two commuting operators on Hilbert spaces and Banach spaces. Results for classes of operators on Hilbert spaces such as hypercontractions in Agler [A2], hyperexpansions in Athavale [At2] and alternating hyperexpansion in Sholapurkar and Athavale [ShT] are obtained by using these formulas. Finally those classes of operators are introduced on Banach spaces.

DOI: 10.14232/actasm-014-550-3

AMS Subject Classification (1991): 47A60, 47A80, 47B99

Keyword(s): isometry, $(m, p)$-isometry, functional calculus, hypercontraction, hyperexpansion, Banach space

Received June 20, 2014, and in revised form April 1, 2015. (Registered under 50/2014.)