Elementary properties of ∞-isometries on a Hilbert space

Inspired by recent works on m-isometries for a positive integer m, in this paper we introduce the classes of ∞-isometries and ∞-unitaries on a Hilbert space. We show that an ∞-isometry on a finite dimensional complex Hilbert space H with dimension N is in fact an (2N−1)-isometry. We describe the spectra of such operators, study the quasinilpotent perturbations of ∞-isometries and characterize when tensor products of ∞-isometries are also ∞-isometries. As a surprising by-product, we obtain a generalization of Nagy-Foias-Langer decomposition of a contraction into an unitary and a completely nonunitary contraction.