@article {15230,
	title = {Onthe Numerical Range and NormofElementary Operators},
	journal = {Linear and Multilinear Algebra},
	volume = {52},
	year = {2004},
	pages = {293-302},
	abstract = {Let B{\dh}E{\TH} be the complex Banach algebra of all bounded linear operators on a complex Banach space E: Forn-tuples A {\textonequarter} {\dh}A1, . . . ,An{\TH} and B {\textonequarter} {\dh}B1, . . . ,Bn{\TH} of operators on E, let RA, B denote the operator on B{\dh}E{\TH}defined by RA, B{\dh}X{\TH} {\textonequarter}Pni{\textonequarter}1AiXBi :For A, B 2 B{\dh}E{\TH}, we put UA, B {\textonequarter} R{\dh}A, B{\TH}, {\dh}B,A{\TH}:In this note, we prove thatcoXni{\textonequarter}1ii : {\dh}1, . . . , n{\TH} 2 V{\dh}A{\TH}, {\dh}1, . . . , n{\TH} 2 V{\dh}B{\TH}( ) W0{\dh}RA, BjJ{\TH}where V{\dh}{\TH} is the joint spatial numerical range, W0{\dh}{\TH} is the algebraic numerical range and J is a norm ideal ofB{\dh}E{\TH}: We shall show that this inclusion becomes an equality when RA, B is taken to be a derivation. Also, wededuce that w{\dh}UA, BjJ{\TH}  2{\dh}ffiffiffi2p 1{\TH}w{\dh}A{\TH}w{\dh}B{\TH}, for A,B 2 B{\dh}E{\TH} and J is a norm ideal of B{\dh}E{\TH}, where w{\dh}{\TH} is thenumerical radius.On the other hand, in the particular case when E is a Hilbert space, we shall prove that the lower estimatebound kUA, BjJk  2{\dh}ffiffiffi2p 1{\TH}kAkkBk holds, if one of the following two conditions is satisfied:(i) J is a standard operator algebra of B{\dh}E{\TH} and A,B 2 J:(ii) J is a norm ideal of B{\dh}E{\TH} and A, B 2 B{\dh}E{\TH}:},
	author = {Seddik. A}
}
