Let $ H $ be a complex Hilbert space and let $ \mathcal{B}(H) $ be the algebra of all bounded linear operators on $ H $. The polar decomposition theorem asserts that every operator $ T \in \mathcal{B}(H) $ can be written as the product $ T = V P $ of a partial isometry $ V\in \mathcal{B}(H) $ and a positive operator $ P \in \mathcal{B}(H) $ such that the kernels of $ V $ and $ P $ coincide. Then this decomposition is unique. $ V $ is called the polar factor of $ T $. Moreover, we have automatically $ P = \vert T\vert = (T^*T)^{\frac{1}{2}} $. Unlike $ P $, we have no representation formula that is required for $ V $.In this paper, we introduce, for $ T\in \mathcal{B}(H) $, a family of functions called a "polar function" for $ T $, such that the polar factor of $ T $ is obtained as a limit of a net built via continuous functional calculus from this family of functions. We derive several explicit formulas representing different polar factors. These formulas allow new for methods of approximations of the polar factor of $ T $.

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希尔伯特空间上算子极因子的表示和近似

令$H$为复数希尔伯特空间，并令$\mathcal{B}(H)$为$H$上所有有界线性算子的代数。极坐标分解定理断言每个算子 $ T \in \mathcal{B}(H) $ 可以写成部分等距的乘积 $ T = VP $ $ V\in \mathcal{B}(H) $ 和一个正运算符 $ P \in \mathcal{B}(H) $ 使得 $ V $ 和 $ P $ 的核重合。那么这个分解是唯一的。$V$被称为$T$的极性因子。此外，我们自动有 $ P = \vert T\vert = (T^*T)^{\frac{1}{2}} $。与$P$不同，我们没有$V$所需要的表示公式。在本文中，我们为$T\in\mathcal{B}(H)$引入了一个称为“极坐标函数”的函数族为 $ T $，这样，$ T $ 的极坐标因子就可以作为通过连续函数演算从该函数族构建的网络的限制而获得的。我们推导出几个表示不同极性因素的明确公式。这些公式为 $ T $ 的极坐标因子的近似方法提供了新的方法。