$$(n_1,\ldots ,n_p)$$(n1,…,np) -quasi- m -isometric commuting tuple of operators on a Hilbert space

Our aim in this paper is to consider a generalization of the concept of n-quasi-m-isometric operators of a single operator done in Mahmoud Sid Ahmed et al. (Results Math 73:511–531, 2018) and Mechri and Mahmoud Sid Ahmed (Oper Matrices 14(1): 145–157, 2020) to the multi-dimensional operators. We discuss the most interesting results concerning these classes of tuples of operators by extending some results obtained in several recent papers for a single operator.