论文标题
标准和数字半径不平等的改进
Refinements of norm and numerical radius inequalities
论文作者
论文摘要
给出了复杂的希尔伯特空间上有限线性算子的标准和数值半径不等式的几种改进。特别是，我们表明，如果$ a $是复杂的希尔伯特空间上的有界线性运算符，则$$ \ frac {1} {4} {4} \ | a^*a^*a+aa+aa^*\ | \ leq \ frac {1} {8} \ big（\ | a+a+a^*\ |^2+\ | a-a^*\ |^2+c^2（a+a^*）+c^2（a-a^*）（a-a^*）\ bigG） \ frac {1} {2} \ | a^*a+aa^*\ | - \ \ frac {1} {4} \ bigG \ |（a+a^*）^2（a-a^*）^2 \ big \ |^{1/2} \ leq w^2（a）\ leq leq \ leq \ leq \ frac \ frac {1}} {1} {2} {2} {2} {2} {2} {2} \ frac {1} {4} \ | a^*a+aa^*\ | \ leq \ frac {1} {2} w^2（a） + \ frac {1} {8} {8} \ bigG \ |（a + a^*）^2（a-a^*）^2 \ big |^big |^|^|^{1/2} {1/2} \ leq w^2（a），$ w^2（a）操作员规范，数值半径和克劳福德号码。此外，我们证明，如果$ a，d $是在复杂的希尔伯特空间上有边界的线性运算符，则\ begin {eqnarray*} \ | ad^*\ | | \ leq \ left \ | \ int_0^1 \ left（（1-t）\ left（\ frac {| a |^2 +| d |^2} {2} {2} {2} \ right） +t \ | ad^*\ | i \ | i \ right）^2dt \ right） | a |^2+| d |^2 \ right \ |，\ end {eqnarray*}其中$ | a | =（a^*a）^{1/2} $和$ | d | =（d^*d^*d）^d）^{1/2} $。这是Bhatia和Kittaneh获得的众所周知的不平等现象的完善。
Several refinements of norm and numerical radius inequalities of bounded linear operators on a complex Hilbert space are given. In particular, we show that if $A$ is a bounded linear operator on a complex Hilbert space, then $$ \frac{1}{4}\|A^*A+AA^*\| \leq \frac{1}{8}\bigg( \|A+A^*\|^2+\|A-A^*\|^2 +c^2(A+A^*)+c^2(A-A^*)\bigg) \leq w^2(A)$$ and \begin{eqnarray*} \frac{1}{2}\|A^*A+AA^*\| - \frac{1}{4}\bigg\|(A+A^*)^2 (A-A^*)^2 \bigg\|^{1/2} \leq w^2(A) \leq \frac{1}{2}\|A^*A+AA^*\|, \end{eqnarray*} %$$ \frac{1}{4}\|A^*A+AA^*\| \leq \frac{1}{2}w^2(A) + \frac{1}{8}\bigg\|(A+A^*)^2 (A-A^*)^2 \bigg\|^{1/2}\leq w^2(A),$$ where $\|.\|$, $w(.)$ and $c(.)$ are the operator norm, the numerical radius and the Crawford number, respectively. Further, we prove that if $A,D$ are bounded linear operators on a complex Hilbert space, then \begin{eqnarray*} \|AD^*\| \leq \left\| \int_0^1 \left( (1-t) \left(\frac{ |A|^2+|D|^2}{2}\right) +t\|AD^*\|I \right)^2dt \right\|^{1/2} \leq \frac{1}{2}\left\| |A|^2+|D|^2 \right\|, \end{eqnarray*} where $|A|=(A^*A)^{1/2}$ and $|D|=(D^*D)^{1/2}$. This is a refinement of well known inequality obtained by Bhatia and Kittaneh.
