论文标题
半希尔伯特太空运营商的一些数值半径不平等
Some numerical radius inequalities for semi-Hilbert space operators
论文作者
论文摘要
令$ a $为积极的线性运算符，在复杂的Hilbert Space $ \ big（\ Mathcal {h}，\ langle \ cdot \ cdot \ cdot \ cdot \ cdot \ rangle \ big）$。令$ω_a（t）$和$ {\ | t \ |} _a $表示$ a $ a $ - numerical radius和$ a $ a $ a-operator eminort $ t $代表在semi-hilbertian space $ \ big big big（\ mathcal { \ cdot \ rangle} _a \ big）$，其中$ {\ langle x \ mid y \ rangle} _a：= \ langle ax \ mid y \ rangle $ for ALL $ x，y \ in \ MATHCAL {H} $。在本文中，我们表明 \ begin {equation*} \ label {m1} \ tfrac {1} {4} \ | t^{\ sharp_a} t+tt^{\ sharp_a} \ | _a \ leω_a^2 \ left（t \ right）\ le \ le \ tfrac {1} {1} t+tt^{\ sharp_a} \ | _a。 \ end {equation*}这里$ t^{\ sharp_a} $表示为$ t $的$ a $ adjoint运算符。此外，证明了上述不平等现象的可观改善。这允许计算操作员矩阵$ \ begin {pmatrix} i＆t \\ 0＆-i \ end end {pmatrix} $ whene $ \ mathbb {a a} = \ text {diag}（a diag}（a，a，a，a，a，a，a）$。此外，还建立了几个$ a $ numerical半径不平等现象，用于半希尔伯特太空运营商。
Let $A$ be a positive bounded linear operator acting on a complex Hilbert space $\big(\mathcal{H}, \langle \cdot\mid \cdot\rangle \big)$. Let $ω_A(T)$ and ${\|T\|}_A$ denote the $A$-numerical radius and the $A$-operator seminorm of an operator $T$ acting on the semi-Hilbertian space $\big(\mathcal{H}, {\langle \cdot\mid \cdot\rangle}_A\big)$ respectively, where ${\langle x\mid y\rangle}_A := \langle Ax\mid y\rangle$ for all $x, y\in\mathcal{H}$. In this paper, we show that \begin{equation*}\label{m1} \tfrac{1}{4}\|T^{\sharp_A} T+TT^{\sharp_A}\|_A\le ω_A^2\left(T\right) \le \tfrac{1}{2}\|T^{\sharp_A} T+TT^{\sharp_A}\|_A. \end{equation*} Here $T^{\sharp_A}$ is denoted to be a distinguished $A$-adjoint operator of $T$. Moreover, a considerable improvement of the above inequalities is proved. This allows to compute the $\mathbb{A}$-numerical radius of the operator matrix $\begin{pmatrix} I&T \\ 0&-I \end{pmatrix}$ where $\mathbb{A}= \text{diag}(A,A)$. In addition, several $A$-numerical radius inequalities for semi-Hilbertian space operators are also established.
