Rank one operators and norm of elementary operators

Ameur Seddik
2007 Linear Algebra and its Applications  
Let A be a standard operator algebra acting on a (real or complex) normed space E. For two n-tuples A = (A 1 , . . . , A n ) and B = (B 1 , . . . , B n ) of elements in A, we define the elementary operator R A,B on A by the relation R A, For a single operator A ∈ A, we define the two particular elementary operators L A and R A on A by L A (X) = AX and R A (X) = XA, for every X in A. We denote by d(R A,B ) the supremum of the norm of R A,B (X) over all unit rank one operators on E. In this note,
more » ... we shall characterize: (i) the supremun d(R A,B ), (ii) the relation d(R A,B ) = n i=1 A i B i , (iii) the relation d(L A − R B ) = A + B , (iv) the relation d(L A R B + L B R A ) = 2 A B . Moreover, we shall show the lower estimate d(L A − R B ) max{sup λ∈V (B) A − λI , sup λ∈V (A) B − λI } (where V (X) is the algebraic numerical range of X in A).
doi:10.1016/j.laa.2006.10.003 fatcat:uu32rsflujconfh6xba4yspese