Let A be a complex normed algebra. For A,B ∈ A, define a basic elementary operator MA,B : A → A by MA,B(X) = AXB. Given a standard operator algebra A acting on a complex normed space and A,B ∈ A we have: (i) The lower estimate kMA,B +MB,Ak ≥ 2(√2 − 1)kAkkBk holds. (ii) The lower estimate kMA,B +MB,Ak ≥ kAkkBk holds if inf ∈C kA + Bk = kAk or inf ∈C kB + Ak = kBk. (iii) The equality kMA,B +MB,Ak = 2kAkkBk holds if kA + Bk = kAk + kBk for some unit scalar . These results extend analogous estimates established earlier for standard operator subalgebras of Hilbert space operators.
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:
