Abstract:

Let BðEÞ be the complex Banach algebra of all bounded linear operators on a complex Banach space E: For
n-tuples A ¼ ðA1, . . . ,AnÞ and B ¼ ðB1, . . . ,BnÞ of operators on E, let RA, B denote the operator on BðEÞ
defined by RA, BðXÞ ¼
Pn
i¼1AiXBi :
For A, B 2 BðEÞ, we put UA, B ¼ RðA, BÞ, ðB,AÞ:
In this note, we prove that
co
Xn
i¼1
ii : ð1, . . . , nÞ 2 VðAÞ, ð1, . . . , nÞ 2 VðBÞ
( )
W0ðRA, BjJÞ
where VðÞ is the joint spatial numerical range, W0ðÞ is the algebraic numerical range and J is a norm ideal of
BðEÞ: We shall show that this inclusion becomes an equality when RA, B is taken to be a derivation. Also, we
deduce that wðUA, BjJÞ 2ð
ffiffiffi2
p
1ÞwðAÞwðBÞ, for A,B 2 BðEÞ and J is a norm ideal of BðEÞ, where wðÞ is the
numerical radius.
On the other hand, in the particular case when E is a Hilbert space, we shall prove that the lower estimate
bound kUA, BjJk 2ð
ffiffiffi2
p
1ÞkAkkBk holds, if one of the following two conditions is satisfied:
(i) J is a standard operator algebra of BðEÞ and A,B 2 J:
(ii) J is a norm ideal of BðEÞ and A, B 2 BðEÞ: