Abstract
We obtain the admissible sets on the unit circle to be the spectrum of a strict m-isometry on an n-finite dimensional Hilbert space. This property gives a better picture of the spectrum of an m-isometry. We determine that the only m-isometries on R2 are 3-isometries and isometries giving by +/- I+Q, where Q is a nilpotent operator. Moreover, on real Hilbert space, we obtain that m-isometries preserve volumes. Also, we present a way to construct a strict (m+1)-isometry with a given m-isometry, using ideas of Aleman and Suciu (Integr Equ Oper Theory 85:259-287, 2016, Proposition 5.2) on infinite dimensional Hilbert space.