论文标题
半希尔伯特太空运营商的广义施瓦茨不平等和一些$ $ numerical radius的不平等现象
The generalized Schwarz inequality for semi-Hilbertian space operators and Some $A$-numerical radius inequalities
论文作者
论文摘要
在这项工作中，证明了半希尔伯特太空运营商的施瓦茨混合不平等。也就是说，对于每一个积极的希尔伯特太空运营商$ a $。如果$ f $和$ g $是$ \ left [0，\ infty \ right）$满足$ f（t）g（t）g（t）= t $ $（t $ $（t \ ge0）$上的非负连续函数，则{\ left \ langle {t x，y} \ right \ rangle_a} \ right | \ le \ left \ | {f \ left（{\ left | t \ right | _a x} \ right）} \ right \ | _a \ left \ | | {g \ left（{\ left | {t^{\ sharp_a}} \ right | _a y} \ right）}} \ right \ | _a \ eend \ eend \ end {align*}对于每个hilbert太空运营商$ t $ a $ t^* a $的范围均与$ a $ a $ a $ a $ a的范围一样$ x，y \ in \ mathscr {h} $，其中$ \ left | t \ right | _a = \ left（at^{\ sharp_a} t \ right）^{1/2} $，以至于$ t^{\ sharp_a} = a^\ dagger t^*a $，其中$ a^\ dagger $是$ a $ a $ a $ a $ a $。基于此，引入了$ A $数字半径的一些不平等。
In this work, the mixed Schwarz inequality for semi-Hilbertian space operators is proved. Namely, for every positive Hilbert space operator $A$. If $f$ and $g$ are nonnegative continuous functions on $\left[0,\infty\right)$ satisfying $f(t)g(t) =t$ $(t\ge0)$, then \begin{align*} \left| {\left\langle {T x,y} \right\rangle_A } \right| \le \left\| {f\left( {\left| T \right|_A x} \right)} \right\|_A \left\| {g\left( {\left| {T^{\sharp_A } } \right|_A y} \right)} \right\|_A \end{align*} for every Hilbert space operator $T$ such that the range of $T^* A$ is a subset in the range of $A$, such that $A$ commutes with $T$, and for all vectors $x,y\in \mathscr{H}$, where $\left| T \right|_A = \left(AT^{\sharp_A}T\right)^{1/2}$ such that $T^{\sharp_A}=A^\dagger T^*A$, where $A^\dagger$ is the Moore-Penrose inverse of $A$. Based on that, some inequalities for the $A$-numerical radius are introduced.
