Let $\mathcal{H}$ be a complex Hilbert space, and A be a positive bounded linear operator on $\mathcal{H}.$ Let ${\mathcal{B}}_A\mathcal{\left(}\mathcal{H}\mathcal{\right)}$ denote the set of all bounded linear operators on $\mathcal{H}$ whose A-adjoint exists. Let $\mathbb{A}$ denote a $2\times 2$ operator matrix of the form $\left[\begin{array}{cc}A& O\\ O& A\end{array}\right]$. Very recently, for a strictly positive operator A, Bhunia et al. [On inequalities for A-numerical radius of operators. Electron J Linear Algebra. 2020;36:143–157] proved an important lemma (Lemma 2.4) to establish several $\mathbb{A}$-numerical radius inequalities for operator matrices in ${\mathcal{B}}_{\mathbb{A}}\mathcal{(}\mathcal{H}⨁\mathcal{H}\mathcal{)}$. In this article, we first prove an analogous result and then provide a new proof of the same lemma by dropping the assumption ‘A is strictly positive’. We then establish several new upper and lower bounds for the $\mathbb{A}$-numerical radius of an operator matrix whose entries are operators in ${\mathcal{B}}_A\mathcal{\left(}\mathcal{H}\mathcal{\right)}.$ Further, we prove some refinements of earlier A-numerical radius inequalities for operators in ${\mathcal{B}}_A\mathcal{\left(}\mathcal{H}\mathcal{\right)}$.

让$\mathcal{H}$是一个复希尔伯特空间，并且A是一个正有界线性算子$\mathcal{H}.$让${\mathcal{乙}}_{\mathrm{一个}}\mathcal{\left(}\mathcal{H}\mathcal{\right)}$表示所有有界线性算子的集合$\mathcal{H}$其A -伴随存在。让$\mathbb{一个}$表示一个$2\times 2$形式的算子矩阵$\left[\begin{array}{cc}\mathrm{一个}& ○\\ ○& \mathrm{一个}\end{array}\right]$. 最近，对于严格的正算子A，Bhunia 等人。[关于算子的A数值半径的不等式。电子 J 线性代数。2020;36:143–157] 被证明是一个重要的引理（引理 2.4），可以建立几个$\mathbb{一个}$-算子矩阵的数值半径不等式${\mathcal{乙}}_{\mathbb{一个}}\mathcal{(}\mathcal{H}⨁\mathcal{H}\mathcal{)}$. 在本文中，我们首先证明了一个类似的结果，然后通过放弃“ A严格为正”的假设来提供相同引理的新证明。然后我们为$\mathbb{一个}$- 一个算子矩阵的数值半径，其条目是算子${\mathcal{乙}}_{\mathrm{一个}}\mathcal{\left(}\mathcal{H}\mathcal{\right)}.$此外，我们证明了早期A数值半径不等式的一些改进${\mathcal{乙}}_{\mathrm{一个}}\mathcal{\left(}\mathcal{H}\mathcal{\right)}$.