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

In this paper we recover an [m,C]-isometric operators and (m,C)-isometric commuting tuples of on a Hilbert space studied respectively in [11] [16], introduce the class [m,C]-isometries for tuple operators. This is generalization spaces. A S = (S1,..., Sp) ? B(H)p said to be p-tuple if ?m (S,C):= j=0 (?1)m?j (m j) (? |?|=j j!/?! CS?CS?)=0 some positive integer m conjugation C. We consider multi-variable these single variable explore their basic properties.