(m,∞)-expansive and (m,∞)-contractive commuting tuple of operators on a Banach space

For a d-tuple of commuting operators S := (S1,..., Sd) ? B[X]d, m N and p (0,?), we define Q(p) (S;u) 0?k?m (-1)k (m k) (???Nd0 |?| = k k!/? ||S?u||p). As natural extension the concepts (m,p)-expansive (m,p)-contractive for tuple operators, introduce study (m,?)-expansive (m,?)-contractive acting on Banach space. We say that is (resp. (m,?)- contractive d-tuple) if Q(p)m 0 u X ?) . These extend definition (m,?)-isometric bounded linear spaces was introduced studied in [13].