In this paper, we give an elementary approach to the numerical radius and norms of the real and imaginary parts of a quadratic operator in terms of its norm. This method is based on proving equality of the numerical radius with one of its suitable upper bounds, via successively establishing equality of the numerical radius with some of its intermediate upper bounds. Meanwhile, some other related results are obtained.

中文翻译：

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二次算子数值半径的一种新方法

在本文中，我们根据范数给出了二次算子实部和虚部的数值半径和范数的基本方法。该方法基于证明数值半径与其合适的上限之一相等，通过连续建立数值半径与其一些中间上限的相等性。同时，还获得了一些其他的相关结果。