论文标题
一对有界线性操作员及其应用的欧几里得操作员半径不平等
Euclidean operator radius inequalities of a pair of bounded linear operators and their applications
论文作者
论文摘要
我们为欧几里得操作员半径的一对有界线性操作员的半径呈尖锐和上边界，这些线性算子在复杂的希尔伯特空间上定义。作为这些界限的应用，我们为有限线性运算符的经典数值半径推导了一系列新界限，该界限对现有的线性运算符的改进。特别是，我们证明，对于有限的线性操作员$ a，$ \ [\ frac {1} {4} \ | a^*a+aa^*\ |+\ |+\ \ \ \ \ \ \ \ \ \ \ \ \ axmax \ {\ | \ | \ re（a）\ |，\ | \ | \ im（a） \ leq w^2（a）\，\ leq \，w^2（| \ re（a）| +i | \ im（a）|），\]，\] 其中$μ= \ big | \ | \ re（a）+\ im（a）\ | - \ | \ re（a） - \ im（a）\ | \ big |。$这改善了数值半径的现有上和下限，即，即 \ [\ frac14 \ | a^*a+aa^*\ | \ leq w^2（a）\ leq \ frac12 \ | a^*a+aa+aa^*\ |。 \]
We obtain several sharp lower and upper bounds for the Euclidean operator radius of a pair of bounded linear operators defined on a complex Hilbert space. As applications of these bounds we deduce a chain of new bounds for the classical numerical radius of a bounded linear operator which improve on the existing ones. In particular, we prove that for a bounded linear operator $A,$ \[\frac{1}{4} \|A^*A+AA^*\|+\fracμ{2}\max \{\|\Re(A)\|,\|\Im(A)\|\} \leq w^2(A) \, \leq \, w^2( |\Re(A)| +i |\Im(A)|),\] where $μ= \big| \|\Re(A)+\Im(A)\|-\|\Re(A)-\Im(A)\|\big|.$ This improve the existing upper and lower bounds of the numerical radius, namely, \[ \frac14 \|A^*A+AA^*\|\leq w^2(A) \leq \frac12 \|A^*A+AA^*\|. \]
