The Norm of Elementary Operators: Numerical Range Bounds and the Hilbert Space Conjecture

This article surveys norm estimates for elementary operators, with particular emphasis on the Jordan elementary operator Ua,b(x) = axb + bxa on ℬ(ℋ). We discuss the established inequality ‖Ua,b‖ ≥ ‖a‖ ‖b‖, which resolves Mathieu’s conjecture, and examine the roles of finite-dimensional reduction, numerical-range geometry and completely bounded norm estimates in its proof. We distinguish numerical-range lower bounds from Haagerup-tensor upper bounds, and we separate the ordinary from the completely bounded norm, noting that the identification of the completely bounded norm with the Haagerup norm on prime C*-algebras does not identify it with the ordinary norm. Explicit matrix examples, verified in full, demonstrate the sharpness of the universal lower constant and illustrate cases of strict inequality. We also discuss extensions to prime C*-algebras and Hilbert modules, and describe the general variational norm formula of Timoney in terms of matrix numerical ranges and the tracial geometric mean.