# Some examples of $m$-isometries

## TL;DR

This paper characterizes the spectra of strict m-isometries on finite-dimensional Hilbert spaces, classifies m-isometries on R^2, and explores volume preservation and construction methods for m-isometries.

## Contribution

It provides a complete description of spectra for strict m-isometries, classifies all m-isometries on R^2, and introduces a construction method for higher-order m-isometries.

## Key findings

- Spectra of strict m-isometries on finite-dimensional spaces are characterized.
- On R^2, m-isometries are limited to 3-isometries and isometries of a specific form.
- m-isometries preserve volume on real Hilbert spaces.

## 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 correct spectrum of an $m$-isometry. We determine that the only $m$-isometries on $\mathbb{R}^2$ are $3$-isometries and isometries giving by $\pm I+Q$, 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 an $m$-isometry given, using ideas of Aleman and Suciu \cite[Proposition 5.2]{AS} on infinite dimensional Hilbert space.