Higher-Order Nonlinear Multivariable Isometric Transformations on Normed Spaces

In this paper, we investigate fundamental properties of (m,p)-isometric tuples ((m,p)−I.T.) in normed spaces. We first establish conditions under which the product of two (m,p)−I.T. remains in the same class, providing a framework for composing such tuples. Next, we derive necessary conditions for an (m,p)−I.T. to become a (2,p)−I.T., characterizing when higher-order isometric behavior can change under parameter adjustments. We also show that any (m,p)−I.T. that is power-bounded reduces to a (1,p)−I.T., revealing a structural collapse from higher-order to first-order isometries under boundedness constraints. Furthermore, we prove that if a tuple (N1,…,Nd) is simultaneously an (m,p)−I.T. and an (m,∞)-isometry, then (N1m,…,Ndm) is a (1,p)−I.T., providing a link between different isometric classes. These results provide a systematic understanding of the stability, transformation behavior, and interrelations of (m,p)−I.T., extending classical operator-theoretic concepts to tuples of commuting linear or nonlinear transformations in normed and Banach spaces.