论文标题
$ \ mathbb {a} $ - 数字半径为$ 2 \ times 2 $ block矩阵的一些上限
Some upper bounds for the $\mathbb{A}$-numerical radius of $2\times 2$ block matrices
论文作者
论文摘要
令$ \ mathbb {a} = \ left（\ begin {array} {cc} a＆0 \\ 0＆a \\ \ end {array} \ right）$是$ 2 \ times2 $ times2 $ diagonal操作员matrix由正限制的操作员确定。对于半希勒伯式运营商$ x $和$ y $，我们首先显示\ begin {align*} w^2 _ {\ mathbb {a}}} \ left（\ begin {bmatrix} 0＆x \\ x \\ y＆0 \ frac {1} {4} \ max \ big \ \ {{{\ big \ | xx^{\ sharp_a} + y^{\ sharp_a} y \ big \ |} _ {a} yy^{\ sharp_a} \ big \ |} _ {a} \ big \} + \ frac {1} {2} {2} \ max \ big \ big \ big \ {w_ {a}（xy） $w_{\mathbb{A}}(\cdot)$, ${\|\cdot\|}_{A}$ and $w_{A}(\cdot)$ are the $\mathbb{A}$-numerical radius, $A$-operator seminorm and $A$-numerical radius, respectively.然后，我们应用上述不平等，以找到$ \ mathbb {a} $ - 某些$ 2 \ times 2 $运算符矩阵的数值半径。特别是，我们获得了早期$ a $ numerical radius的一些改进，用于半希尔伯特式运营商。还给出了$ \ mathbb {a} $ - $ 2 \ times 2 $ block block矩阵的$ \ mathbb {a} $的上限。
Let $\mathbb{A}=\left( \begin{array}{cc} A & 0 \\ 0 & A \\ \end{array} \right)$ be the $2\times2$ diagonal operator matrix determined by a positive bounded operator $A$. For semi-Hilbertian operators $X$ and $Y$, we first show that \begin{align*} w^2_{\mathbb{A}}\left(\begin{bmatrix} 0 & X \\ Y & 0 \end{bmatrix}\right) &\leq \frac{1}{4}\max\Big\{{\big\|XX^{\sharp_A} + Y^{\sharp_A}Y\big\|}_{A}, {\big\|X^{\sharp_A}X + YY^{\sharp_A}\big\|}_{A}\Big\} + \frac{1}{2}\max\big\{w_{A}(XY), w_{A}(YX)\big\}, \end{align*} where $w_{\mathbb{A}}(\cdot)$, ${\|\cdot\|}_{A}$ and $w_{A}(\cdot)$ are the $\mathbb{A}$-numerical radius, $A$-operator seminorm and $A$-numerical radius, respectively. We then apply the above inequality to find some upper bounds for the $\mathbb{A}$-numerical radius of certain $2\times 2$ operator matrices. In particular, we obtain some refinements of earlier $A$-numerical radius inequalities for semi-Hilbertian operators. An upper bound for the $\mathbb{A}$-numerical radius of $2\times 2$ block matrices of semi-Hilbertian space operators is also given.
