Abstract :
Let image 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 image, we define the elementary operator RA,B on image by the relation image for all X in image. For a single operator image, we define the two particular elementary operators LA and RA on image by LA(X) = AX and RA(X) = XA, for every X in image. 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 image, (iii) the relation d(LA − RB) = short parallelAshort parallel + short parallelBshort parallel, (iv) the relation d(LARB − LBRA) = 2short parallelAshort parallel + short parallelBshort parallel. Moreover, we shall show the lower estimate d(LA − RB) greater-or-equal, slanted max{supλset membership, variantV(B)short parallelA − λIshort parallel, supλset membership, variantV(A)short parallelB − λIshort parallel} (where V(X) is the algebraic numerical range of X in image).
