Weighted shift and composition operators on ℓp which are (m,q)-isometries

We obtain some conditions about the weighted shift and the composition operators on ℓpℓp to be (m,q)(m,q)-isometries. In particular, we prove that if T   is a strict (2,q)(2,q) isometry, then p=qp=q, whether T   is a weighted shift or a composition operator on ℓpℓp. Moreover, we characterize the admissible weights for the weighted shift to be a strict (3,p)(3,p)-isometry on ℓpℓp. The case of strict (4,p)(4,p)-isometry is also studied.