Abstract
For a d-tuple of commuting operators S := (S-1, ..., S-d) is an element of B[X](d), m is an element of N and p is an element of (0, infinity), we define
Q(m)((p)) (S; u) := Sigma(0 <= k <= m) (-1)(k)((m)(k))(Sigma(mu) (is an element of) (N0d) (vertical bar mu vertical bar = k) k!/mu parallel to S(mu)u parallel to(p)).
As a natural extension of the concepts of (m, p)-expansive and (m, p)-contractive for tuple of commuting operators, we introduce and study the concepts of (m, infinity)-expansive tuple and (m, infinity)-contractive tuple of commuting operators acting on a Banach space. We say that S is (m, infinity)-expansive d-tuple (resp. (m, infinity)- contractive d-tuple ) of operators if Q(m)((p))(S; u) <= 0 for all u is an element of X and p -> infinity (resp. Q(m)((p)) (S; u) >= 0 for all u is an element of X and p -> infinity). These concepts extend the definition of (m, infinity)-isometric tuple of bounded linear operators acting on Banach spaces was introduced and studied in [13].