Higher dimensional [m,C]-isometric commuting d-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 [11] and [16], we introduce the class of [m,C]-isometries for tuple of commuting operators. This is a generalization of the class of [m,C]-isometric commuting operators on a Hilbert spaces. A commuting tuples of operators S = (S1,..., Sp) ? B(H)p is said to be [m,C]-isometric p-tuple of commuting operators if ?m (S,C):= ?m j=0 (?1)m?j (m j) (? |?|=j j!/?! CS?CS?)=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.