论文标题
改善了与$ a $ numerical半径相关的不平等现象
Improved inequalities related to the $A$-numerical radius for commutators of operators
论文作者
论文摘要
让$ a $为复杂的希尔伯特太空$ \ Mathcal {h} $和$ \ Mathcal {b} _ {a}（\ Mathcal {h}）$上的正界线性操作员是所有运营商的子空间。在本文中，我们建立了一些不平等，涉及换向者和抗抗议机在半山地空间中，即是由阳性半芬酸盐形式产生的空间。主要是，除其他不等式外，我们证明，对于$ t，s \ in \ Mathcal {b} _ {a}（\ Mathcal {h}）$，我们有\ begin \ begin {align*}ω_a（ts \ p pm st） f_a（s，t）\ big \}，\ end {align*}其中$ f_a（x，y）= \ | y \ | _a \ sqrt {ω_a^2（x） - \ frac {\ left | \，\ weled \ | \ | \ frac {x+x+x^{\ sharp_a}}} {2} \ right \ | _a^2- \左\ | \ | \ frac {x-x^{\ sharp_a}}} {2i} \ right \ | | _a^2 \ right |} {2} {2}} {2}}}}。$$ $ a $ a $ a $ numerical半径和$ a $ a $ a-hilbert太空运营商的$ a $ a-numerical半径和$ x^{\ sharp_a} $表示$ x^{$ x^{\ a sharp_a} $表示，$ a $ a $ a $ x $ x $ x $。
Let $A$ be a positive bounded linear operator on a complex Hilbert space $\mathcal{H}$ and $\mathcal{B}_{A}(\mathcal{H})$ be the subspace of all operators which admit $A$-adjoints operators. In this paper, we establish some inequalities involving the commutator and the anticommutator of operators in semi-Hilbert spaces, i.e. spaces generated by positive semidefinite sesquilinear forms. Mainly, among other inequalities, we prove that for $T, S\in\mathcal{B}_{A}(\mathcal{H})$ we have \begin{align*} ω_A(TS \pm ST) \leq 2\sqrt{2}\min\Big\{f_A(T,S), f_A(S,T) \Big\}, \end{align*} where $$f_A(X,Y)=\|Y\|_A\sqrt{ω_A^2(X)-\frac{\left|\,\left\|\frac{X+X^{\sharp_A}}{2}\right\|_A^2-\left\|\frac{X-X^{\sharp_A}}{2i}\right\|_A^2\right|}{2}}.$$ Here $ω_A(\cdot)$ and $\|\cdot\|_A$ are the $A$-numerical radius and the $A$-operator seminorm of semi-Hilbert space operators, respectively and $X^{\sharp_A}$ denotes a distinguished $A$-adjoint operator of $X$.
