Abstract
The study of tuples of commuting operators was the subject of intensive study by many authors. Our aim in this work is to consider a generalization of the notions of m-partial isometries and (m, q)-partial isometries (resp. m- left inverse and m-right inverse) of a single operator done in [23] and [21] (resp. in [14],[19], [22]) to the multivariable operators. We study some of the basic properties of these tuples of commuting operators. A commuting d-tuple of operators T = (T-1, ..., T-d) acting on a Hilbert space H is called a joint (m, (q(1), ..., q(d)))-partial isometry, if
Tq(Sigma(0 <= k <= m) (-1)(k)(GRAPHICS)Sigma(vertical bar alpha vertical bar=k) k!/alpha! T*T-alpha(alpha)) = 0.