Strict isometries of arbitrary orders

We consider the elementary operator L, acting on the Hilbert–Schmidt class C2(H), given by L(T)=ATB, with A and B bounded operators on a separable Hilbert space H. In this paper we establish results relating isometric properties of L with those of the defining symbols A and B. We also show that if A is a strict n-isometry on a Hilbert space H then is a linearly independent set of operators. This result allows to extend further the isometric interdependence of L and its symbols. In particular we show that if L is a p-isometry then A is a strict p-1- (or p-2-)isometry if and only if B∗ is a strict 2-(or 3-)isometry.