<?xml version="1.0" encoding="UTF-8"?><xml><records><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Seddik. A</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">Rank one operators and norm of elementary operators</style></title><secondary-title><style face="normal" font="default" size="100%">Linear Algebra Appl.</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2007</style></year></dates><volume><style face="normal" font="default" size="100%">424</style></volume><pages><style face="normal" font="default" size="100%">177-183</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">Let A be a standard operator algebra acting on a (real or complex) normed space E. For two n-tuples&lt;br&gt;A = (A1, . . . , An) and B = (B1, . . . , Bn) of elements in A, we define the elementary operator RA,B on&lt;br&gt;A by the relation RA,B(X) = n&lt;br&gt;i=1 AiXBi for all X in A. For a single operator A ∈ A, we define the&lt;br&gt;two particular elementary operators LA and RA onAby LA(X) = AX and RA(X) = XA, for every X in&lt;br&gt;A. We denote by d(RA,B) the supremum of the norm of RA,B(X) over all unit rank one operators on E.&lt;br&gt;In this note, we shall characterize: (i) the supremun d(RA,B), (ii) the relation d(RA,B) = n&lt;br&gt;i=1&lt;br&gt;Ai&lt;br&gt;Bi&lt;br&gt;,&lt;br&gt;(iii) the relation d(LA&lt;br&gt;− RB) = A + B, (iv) the relation d(LARB&lt;br&gt;+ LBRA) = 2AB. Moreover,&lt;br&gt;we shall showthe lower estimate d(LA&lt;br&gt;− RB)  max{supλ∈V (B)&lt;br&gt;A − λI, supλ∈V (A)&lt;br&gt;B − λI} (where&lt;br&gt;V (X) is the algebraic numerical range of X inA).</style></abstract></record></records></xml>