Products of m-isometries

An operator T on a Banach space X is called an (m,p)-isometry if it satisfies the equality ∑k=0mmk(-1)m-k‖Tkx‖p=0, for all x∈X. In this paper we prove that if T is an (n,p)-isometry, S is an (m,p)-isometry and they commute, then TS is an (m+n-1,p)-isometry. This result applied to elementary operators of length 1 defined on the Hilbert-Schmidt class proves a conjecture in [11].