ABSTRACT

Let $\mathcal{B}\mathcal{(}\mathcal{H}\mathcal{)}$ denote the algebra of all bounded linear operators acting on a complex Hilbert space $\mathcal{H}$. For $A,B\in \mathcal{B}\mathcal{(}\mathcal{H}\mathcal{)}$, define the bimultiplication operator ${\mathit{M}}_{\mathit{2}\mathit{,}\mathit{A}\mathit{,}\mathit{B}}$ on the class of Hilbert–Schmidt operators by ${\mathit{M}}_{\mathit{2}\mathit{,}\mathit{A}\mathit{,}\mathit{B}}\mathit{(}\mathit{X}\mathit{)}\mathit{=}\mathit{A}\mathit{X}\mathit{B}$. In this paper, we show that if B is normal, then $co({\mathit{W}}_0(A){\mathit{W}}_0(B))\subseteq W_0({\mathit{M}}_{\mathit{2}\mathit{,}\mathit{A}\mathit{,}\mathit{B}}\mathit{)}\mathit{,}$where co stands for the convex hull and $W_0(.)$ denotes the maximal numerical range. If in addition, A is hyponormal, this inclusion becomes an equality. Some remarks about the maximal numerical range of the generalized derivation ${\delta }_{2,A,B}$ on the class of Hilbert–Schmidt operators are also given.

中文翻译：

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关于二乘法M2,A,B的最大数值范围

摘要

让$\mathcal{乙}\mathcal{(}\mathcal{H}\mathcal{)}$表示作用于复数希尔伯特空间的所有有界线性算子的代数$\mathcal{H}$. 为了$\mathrm{一种},乙\in \mathcal{乙}\mathcal{(}\mathcal{H}\mathcal{)}$, 定义双乘运算符${\mathit{米}}_{\mathit{2个}\mathit{,}\mathit{一种}\mathit{,}\mathit{乙}}$在 Hilbert–Schmidt 算子类上${\mathit{米}}_{\mathit{2个}\mathit{,}\mathit{一种}\mathit{,}\mathit{乙}}\mathit{(}\mathit{X}\mathit{)}\mathit{=}\mathit{一种}\mathit{X}\mathit{乙}$. 在本文中，我们证明如果B是正规的，则$Co({\mathit{W}}_0(\mathrm{一种}){\mathit{W}}_0(乙))\subseteq W_0({\mathit{米}}_{\mathit{2个}\mathit{,}\mathit{一种}\mathit{,}\mathit{乙}}\mathit{)}\mathit{,}$其中co代表凸包和$W_0(.)$表示最大数值范围。此外，如果A是次正规的，则此包含变为相等。关于广义推导的最大数值范围的一些说明${\delta }_{\mathrm{2个},\mathrm{一种},乙}$还给出了 Hilbert–Schmidt 算子类。