论文标题
半希尔伯特空间操作员的半径和数值半径不平等的细化
Refinement of seminorm and numerical radius inequalities of semi-Hilbertian space operators
论文作者
论文摘要
我们以$ $ a-operator eminorm和$ a $ numerical radius的半希尔伯特太空运营商的半径给予新的不平等，并表明此处获得的不平等现象对现有的不平等程度进行了概括和改善。考虑一个复杂的Hilbert Space $ \ Mathcal {H} $和$ \ Mathcal {h}上的非零正界线性运算符$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ in \ s，t，x \ in \ s，t，x \ in \ mathcal ro. $ s，t，x $存在然后$$ 2 \ | s^{\ sharp_a} Xt xt \ | _a \ leq \ leq \ | ss^{\ sharp_a} x+x+x+x+xtt^{\ sharp_a} \ | _a。$ _a。 \ begin {eqnarray*} \ frac {1} {4} \ | t^{\ sharp_ {a}} t+tt+tt^{\ sharp_ {a}}} \ | _a \ | _a \ leq \ leq \ leq \ frac \ frac {1} {1} {8} {8} \ big big big big（ \ | t+t^{\ sharp_ {a}} \ | _a^2+\ | | t-t-t^{\ sharp_ {a}} \ | _a^2 \ bigg），~~ \ textit {and} \ | t+t^{\ sharp_ {a}}} \ | _a^2+\ | t-t-t-t^{\ sharp_ {a}} \ | _a^2 \ bigG） +\ frac {1} {8} c_a^2 \ big（t+t^{\ sharp_ {a}} \ big）+\ frac {1} {8} c_a^2 \ big（t-t-t^t^{\ sharp_ {a}}}}}} \ big）\ big） \ end {eqnarray*}在这里$ w_a（。），c_a（。）$和$ \ |。|。\ | _a $ deote $ a $ a $ numerical radius，$ a $ a $ -crawford编号和$ a $ a $ a-operator eminorm。
We give new inequalities for $A$-operator seminorm and $A$-numerical radius of semi-Hilbertian space operators and show that the inequalities obtained here generalize and improve on the existing ones. Considering a complex Hilbert space $\mathcal{H}$ and a non-zero positive bounded linear operator $A$ on $\mathcal{H},$ we show with among other seminorm inequalities, if $S,T,X\in \mathcal{B}_A(\mathcal{H})$, i.e., if $A$-adjoint of $S,T,X$ exist then $$2\|S^{\sharp_A}XT\|_A \leq \|SS^{\sharp_A}X+XTT^{\sharp_A}\|_A.$$ Further, we prove that if $T\in \mathcal{B}_A(\mathcal{H})$ then \begin{eqnarray*} \frac{1}{4}\|T^{\sharp_{A}}T+TT^{\sharp_{A}}\|_A \leq \frac{1}{8}\bigg( \|T+T^{\sharp_{A}}\|_A^2+\|T-T^{\sharp_{A}}\|_A^2\bigg), ~~\textit{and} \end{eqnarray*} \begin{eqnarray*} \frac{1}{8}\bigg( \|T+T^{\sharp_{A}}\|_A^2+\|T-T^{\sharp_{A}}\|_A^2\bigg) +\frac{1}{8}c_A^2\big(T+T^{\sharp_{A}}\big)+\frac{1}{8}c_A^2\big(T-T^{\sharp_{A}}\big) \leq w^2_A(T). \end{eqnarray*} Here $w_A(.), c_A(.)$ and $\|.\|_A $ denote $A$-numerical radius, $A$-Crawford number and $A$-operator seminorm, respectively.
