Abstract. In this paper we consider the notion of $n$-isometry on a Banach space. We determine the existence of non-isometric $2$-isometries for the classes of weighted shifts and composition operators on $\ell_p$ spaces. We also address similar questions for weighted composition operators acting on the space of scalar valued continuous functions, defined on a compact Hausdorff topological space.

AMS Subject Classification (1991): 47B33, 47B37

Keyword(s): $n$-isometries, weighted composition operators on $l_p$ spaces, composition operators on ${\cal C}(X)$

Received January 12, 2009, and in revised form July 7, 2009. (Registered under 5/2009.)