Functional calculus for m-isometries and related operators on Hilbert spaces and Banach spaces

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 [13] where it was proved that if T is an (m, p)-isometry on a Banach space, then Tr 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 [1], hyperexpansions in Athavale [7] and alternating hyperexpansion in Sholapurkar and Athavale [41] are obtained by using these formulas. Finally those classes of operators are introduced on Banach spaces. © 2015 Bolyai Institute, University of Szeged.