Higher dimensional $[m,C]$-isometric commuting $p$-tuple of operators

Abstract

In this paper we recover an $[m, C]$-isometric operators and (m, C)-isometric commuting tuples of operators on a Hilbert space studied respectively in \cite{CKL2} and \cite{SCL1}, we introduce the class of [m,C]-isometries for tuple of commuting operators. This is a generalization of the class of $[m, C]$-isometric operators on a Hilbert spaces. A commuting tuples of operators
${\bf \large S}=(S_1,\cdots,S_p)\in \mathcal{L}(\mathcal{H})^p$ is said to be $[m,C]$-isometric $p$-tuple of commuting operators if
$${\Psi}_{m}\big( \;{\bf\large S}, C \big):=\sum_{j=0}^{ m}(-1)^{m-j}\binom{m}{j}\bigg(\sum_{|\alpha|=j}\frac{j!}{\alpha!}C{\bf \large S}^{\alpha}C{\bf\large S}^{\alpha}\bigg)=0$$ for some positive integer $m$ and some conjugation $C$. We consider a multi-variable generalization of these single variable $[m, C]$-isometric operators and explore some of their basic properties.