On (m,∞)-isometries: Examples

An operator T on a Banach space X is said to be an ( m , ∞ ) -isometry, if max 0 ≤ k ≤ m k even ‖ T k x ‖ = max 0 ≤ k ≤ m k odd ‖ T k x ‖ , for all x ∈ X . In this paper, we study unilateral weighted shift operators which are ( m , ∞ ) -isometries for some integers m . In particular, we show that any power of an ( m , ∞ ) -isometry is not necessarily an ( m , ∞ ) -isometry. We also study strict ( 3 , ∞ ) -isometries on R 2 and give an example of a strict ( 2 n - 1 , ∞ ) -isometry on C 2 , for any odd integer n .