Elementary operators which are m-isometries

The study of m-isometry elementary operator of length one is initiated by Botelho and Jamison. Several authors have contributed to obtain this result: if the two operator coefficients involved are m-isometries then the elementary operator is m-isometry on the Hilbert-Schmidt operator ideal. In this paper we show that the converse of this result is also true. We also give a necessary and sufficient condition for the generalized derivation to be an m-isometry. The sufficiency follows from results on the sum of m-isometries and nilpotent operators in two recent papers.