Strict isometries of arbitrary orders

We consider the elementary operator L, acting on the Hilbert-Schmidt class C-2(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*)(2)A(2)..., (A*)(n-1)A(n-1)} is a linearly independent set of operators. This result allows to extend further the isometric interdependence of C 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. (C) 2011 Elsevier Inc. All rights reserved.