论文标题
两家运营商及其应用
Inequalities for the $A$-joint numerical radius of two operators and their applications
论文作者
论文摘要
令$ \ big（\ Mathcal {h}，\ langle \ cdot \ cdot \ cdot \ rangle \ big）$为复杂的希尔伯特空间，$ a $ a $是$ \ \ \ nathcal {h} $的正面（半罚款）有界线性操作员。由$ a $诱导的半inner产品由$ {\ langle x \ mid y \ rangle} _a：= \ langle ax \ mid y \ rangle $，$ x，y in \ in \ nathcal {h h} $，并定义了一个seminorm $ $ {\ cdot \ cdot \ c $ c $这使$ \ Mathcal {h} $进入半希尔伯特空间。 $ a $ - 连接数值半径为两个$ a $ a $ bound的运算符$ t $和$ s $由\ begin {align*}ω__{a，\ text {e}}（t，t，t，s）= \ sup _ { x \ rangle} _a \ big |^2+\ big | {\ langle sx \ mid x \ rangle} _a _a \ big |^2}。 \ end {align*}在本文中，我们的目标是证明涉及$ω__{a，\ text {e}}}（t，s）$的几个范围。此外，还建立了与$ -Davis-Wielandt半径相关的半希尔伯特太空运营商。一些获得的边界概括并完善了Zamani和Shebrawi的一些早期结果[Mediterr。 J. Math。 17，25（2020）]。
Let $\big(\mathcal{H}, \langle \cdot\mid \cdot\rangle \big)$ be a complex Hilbert space and $A$ be a positive (semidefinite) bounded linear operator on $\mathcal{H}$. The semi-inner product induced by $A$ is given by ${\langle x\mid y\rangle}_A := \langle Ax\mid y\rangle$, $x, y\in\mathcal{H}$ and defines a seminorm ${\|\cdot\|}_A$ on $\mathcal{H}$. This makes $\mathcal{H}$ into a semi-Hilbert space. The $A$-joint numerical radius of two $A$-bounded operators $T$ and $S$ is given by \begin{align*} ω_{A,\text{e}}(T,S) = \sup_{\|x\|_A= 1}\sqrt{\big|{\langle Tx\mid x\rangle}_A\big|^2+\big|{\langle Sx\mid x\rangle}_A\big|^2}. \end{align*} In this paper, we aim to prove several bounds involving $ω_{A,\text{e}}(T,S)$. Moreover, several inequalities related to the $A$-Davis-Wielandt radius of semi-Hilbert space operators is established. Some of the obtained bounds generalize and refine some earlier results of Zamani and Shebrawi [Mediterr. J. Math. 17, 25 (2020)].
