Norms and CB norms of Jordan elementary operators

We establish lower bounds for norms and CB-norms of elementary operators on B(H). Our main result concerns the operator T(a,b)x = axb + bxa and we show parallel toT(a,b)parallel to greater than or equal to parallel toaparallel to parallel tobparallel to, proving a conjecture of M. Mathieu. We also establish some other results and formulae for parallel toT(a,b)parallel to(cb) and parallel toT(a,b)parallel to for special cases. (C) 2003 Editions scientifiques et medicales Elsevier SAS. All rights reserved.