Every expansive $ m $-concave operator has $ m $-isometric dilation

TL;DR

This paper establishes the existence of minimal $m$-isometric dilations for expansive $m$-concave operators on Hilbert spaces, providing explicit matrix representations and exploring special cases where expansivity is unnecessary.

Contribution

It introduces a method to obtain minimal $m$-isometric dilations for expansive $m$-concave operators, including cases with relaxed assumptions.

Findings

Minimal $m$-isometric dilations exist for expansive $m$-concave operators.

Explicit matrix representations of the dilations are provided.

In the case of 3-concave operators, expansivity assumptions are unnecessary.

Abstract

The aim of this paper is to obtain $ m $-isometric dilation of expansive $ m $-concave operator on Hilbert space. The obtained dilation is shown to be minimal. The matrix representation of this dilation is given. It is also proved that in case of 3-concave operators the assumption on expansivity is not necessary. The paper contains an example showing that minimal $ m $-isometric dilations may not be isomorphic.