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 Ta,bx = axb + bxa and we show ‖Ta,b‖ ≥ ‖a‖‖b‖, proving a conjecture of M. Mathieu. We also establish some other results and formulae for ‖Ta,b‖cb and ‖Ta,b‖ for special cases. Our results are related to a problem of M. Mathieu [13, 14] asking whether ‖Ta,b‖ ≥ c‖a‖‖b‖ holds in general with c = 1. We prove this in Theorem 6 below. In [14] the inequality is established for c = 2/3 and the best known result to date is c = 2( √ 2− 1) as shown in [17, 5, 11]. There are simple examples which show that c cannot be greater than 1 in general and there are results which prove the inequality with c = 1 in special cases. The case a = a and b = b is shown in [12] where it is deduced from ‖Ta,b‖cb = ‖Ta,b‖ under these hypotheses. The equality of the the CB norm and the operator norm of Ta,b also holds if a, b are commuting normal operators. See section 3 below for references. A result for c = 1 is shown in [2] under the assumption that ‖a+zb‖ ≥ ‖a‖ for all z ∈ C. In more general contexts similar results (with varying values of c) are shown in [6, 5]. As this manuscript was being written we learned of another proof of the main result ([4]), using rather different methods. Thanks are due to M. Mathieu for drawing our attention to this reference. Acknowledgement. Part of this work was done during a visit by the author to the University of Edinburgh in the autumn of 2002. A significant impetus to the work arose from discussions with Bojan Magajna and Aleksej Turnšek during a visit to Ljubljana in March 2003 and the author is very grateful to them for that. Thanks also to P. Legǐsa for finding the reference to [10] below. 1 1 Preliminaries We call T :B(H) → B(H) an elementary operator if T has a representation T (x) = l ∑ i=1 aixbi with ai, bi ∈ B(H) for each i. We cite [1] for an exposition of many of the known results on (more general) elementary operators and for other concepts we cite a number of treatises on operator spaces including [8, 15, 7]. In particular we will use the completely bounded (or CB) norm ‖T‖cb of an elementary operator, the operator norm ‖T‖ and the estimate in terms of the Haagerup tensor product norm ‖T‖ ≤ ‖T‖cb ≤ ∥