论文标题
Hilbert-Schmidt Norm的几何方法
A geometric approach to inequalities for the Hilbert--Schmidt norm
论文作者
论文摘要
我们在非零hilbert-schmidt操作员$ x $和$ y $之间定义角度$θ_{x，y} $ by $ \cosθ_ {_ {x，x，y}} = \ frac {{\ rm re} {\ rm re} {\ rm rm tr}（y^*x）} {{{\ | x \ |} _ {_ 2} {\ | y \ |} _ {_ _ 2}} $，并给出一些本质上的属性。除其他外，\ begin {align*} \ big | \cosθ_ {_ {_ {x，y}}} \ big | \ leq \ leq \ min \ min \ left \ left \ sqrt {\ sqrt {\cosθ_ {_ {_ {_ {_ {_ {_ { \ sqrt {\cosθ_ {_ {| x |，| y |}}}}} \ right \}。 \ end {Align*}它使我们能够为希尔伯特 - schmidt Norm提供一些众所周知的不平等现象。特别是，我们应用这种不等式来证明李的猜想[线性代数应用。 433（2010），编号〜3，580--584]如下\ begin {align*} {\ big \ | x + y \ y \ big \ |} _ {_ {_ _ _ _ _ _ {_ _ leq \ leq \ sqrt { + | y | \，\ big \ |} _ {_ 2}。 \ end {align*}提出了一个数值示例，以显示常数$ \ sqrt {\ frac {\ sqrt {\ sqrt {2} + 1} {2}}} $是最小的。还考虑了希尔伯特 - 希尔伯特 - 希尔伯特规范的其他相关不平等。
We define angle $Θ_{X,Y}$ between non-zero Hilbert--Schmidt operators $X$ and $Y$ by $\cosΘ_{_{X,Y}} = \frac{{\rm Re}{\rm Tr}(Y^*X)}{{\|X\|}_{_2}{\|Y\|}_{_2}}$, and give some of its essentially properties. It is shown, among other things, that \begin{align*} \big|\cosΘ_{_{X,Y}}\big|\leq \min\left\{\sqrt{\cosΘ_{_{|X^*|,|Y^*|}}}, \sqrt{\cosΘ_{_{|X|,|Y|}}}\right\}. \end{align*} It enables us to provide alternative proof of some well-known inequalities for the Hilbert--Schmidt norm. In particular, we apply this inequality to prove Lee's conjecture [Linear Algebra Appl. 433 (2010), no.~3, 580--584] as follows \begin{align*} {\big\|X + Y\big\|}_{_2} \leq \sqrt{\frac{\sqrt{2} + 1}{2}}\,{\big\|\,|X| + |Y|\,\big\|}_{_2}. \end{align*} A numerical example is presented to show the constant $\sqrt{\frac{\sqrt{2} + 1}{2}}$ is smallest possible. Other related inequalities for the Hilbert--Schmidt norm are also considered.
