Strict Isometries of Arbitray 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 {I, A∗A, (A∗)2A2, . . . , (A∗)n−1An−1} is a linearly independent set of operators. This results allows to extend further the isometric interdependence of L and its symbols. In particular we show that if L is an 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.