Abstract:

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 RA,B(X) = n
i=1 AiXBi for all X in A. For a single operator A ∈ A, we define the
two particular elementary operators LA and RA onAby 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 d(RA,B) = n
i=1
Ai
Bi
,
(iii) the relation d(LA
− RB) = A + B, (iv) the relation d(LARB
+ LBRA) = 2AB. Moreover,
we shall showthe 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 inA).