If A, B are bounded linear operators on a complex Hilbert space, then we prove that

$$\begin{aligned} w(A)\le & {} \frac{1}{2}\left( \Vert A\Vert +\sqrt{r\left( |A||A^*|\right) }\right) ,\\ w(AB \pm BA)\le & {} 2\sqrt{2}\Vert B\Vert \sqrt{ w^2(A)-\frac{c^2(\mathfrak {R}(A))+c^2(\mathfrak {I}(A))}{2} }, \end{aligned}$$

where \(w(\cdot ),\left\| \cdot \right\| \), and \(r(\cdot )\) are the numerical radius, the operator norm, the Crawford number, and the spectral radius respectively, and \(\mathfrak {R}(A)\), \(\mathfrak {I}(A)\) are the real part, the imaginary part of A respectively. The inequalities obtained here generalize and improve on the existing well known inequalities.

中文翻译：

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希尔伯特空间算子数值半径不等式的推进

如果A ,  B是复 Hilbert 空间上的有界线性算子，那么我们证明

$$\begin{aligned} w(A)\le & {} \frac{1}{2}\left( \Vert A\Vert +\sqrt{r\left( |A||A^*|\right ) }\right) ,\\ w(AB \pm BA)\le & {} 2\sqrt{2}\Vert B\Vert \sqrt{ w^2(A)-\frac{c^2(\mathfrak {R}(A))+c^2(\mathfrak {I}(A))}{2} }, \end{aligned}$$

其中\(w(\cdot ),\left\| \cdot \right\| \)和\(r(\cdot )\)分别是数值半径、算子范数、克劳福德数和谱半径和\（\ mathfrak {R}（A）\） ，\（\ mathfrak {I}（A）\）是实部，虚部甲分别。这里获得的不等式概括并改进了现有的众所周知的不等式。