<?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%">Onthe Numerical Range and NormofElementary Operators</style></title><secondary-title><style face="normal" font="default" size="100%">Linear and Multilinear Algebra</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2004</style></year></dates><volume><style face="normal" font="default" size="100%">52</style></volume><pages><style face="normal" font="default" size="100%">293-302</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">Let BðEÞ be the complex Banach algebra of all bounded linear operators on a complex Banach space E: For&lt;br&gt;n-tuples A ¼ ðA1, . . . ,AnÞ and B ¼ ðB1, . . . ,BnÞ of operators on E, let RA, B denote the operator on BðEÞ&lt;br&gt;defined by RA, BðXÞ ¼&lt;br&gt;Pn&lt;br&gt;i¼1AiXBi :&lt;br&gt;For A, B 2 BðEÞ, we put UA, B ¼ RðA, BÞ, ðB,AÞ:&lt;br&gt;In this note, we prove that&lt;br&gt;co&lt;br&gt;Xn&lt;br&gt;i¼1&lt;br&gt;ii : ð1, . . . , nÞ 2 VðAÞ, ð1, . . . , nÞ 2 VðBÞ&lt;br&gt;( )&lt;br&gt; W0ðRA, BjJÞ&lt;br&gt;where VðÞ is the joint spatial numerical range, W0ðÞ is the algebraic numerical range and J is a norm ideal of&lt;br&gt;BðEÞ: We shall show that this inclusion becomes an equality when RA, B is taken to be a derivation. Also, we&lt;br&gt;deduce that wðUA, BjJÞ  2ð&lt;br&gt;ffiffiffi2&lt;br&gt;p&lt;br&gt; 1ÞwðAÞwðBÞ, for A,B 2 BðEÞ and J is a norm ideal of BðEÞ, where wðÞ is the&lt;br&gt;numerical radius.&lt;br&gt;On the other hand, in the particular case when E is a Hilbert space, we shall prove that the lower estimate&lt;br&gt;bound kUA, BjJk  2ð&lt;br&gt;ffiffiffi2&lt;br&gt;p&lt;br&gt; 1ÞkAkkBk holds, if one of the following two conditions is satisfied:&lt;br&gt;(i) J is a standard operator algebra of BðEÞ and A,B 2 J:&lt;br&gt;(ii) J is a norm ideal of BðEÞ and A, B 2 BðEÞ:</style></abstract></record></records></xml>