Rank one operators and norm of elementary operators

Let A be a standard operator algebra acting on a (real or complex) normed space E. For two n-tuples A = (A1, … , An) and B = (B1, … , Bn) of elements in A, we define the elementary operator RA,B on A by the relation for all X in A. For a single operator A∈A, we define the two particular elementary operators LA and RA on A by LA(X) = AX and RA(X) = XA, for every X in A. We denote by d(RA,B) the supremum of the norm of RA,B(X) over all unit rank one operators on E. In this note, we shall characterize: (i) the supremun d(RA,B), (ii) the relation , (iii) the relation d(LA − RB) = ∥A∥ + ∥B∥, (iv) the relation d(LARB − LBRA) = 2∥A∥ + ∥B∥. Moreover, we shall show the lower estimate d(LA − RB) ⩾ max{supλ∈V(B)∥A − λI∥, supλ∈V(A)∥B − λI∥} (where V(X) is the algebraic numerical range of X in A).