We revise Krein's extension theory of positive symmetric operators. Our approach using factorization through an auxiliary Hilbert space has several advantages: it can be applied to non-densely defined transformations and it works in both real and complex spaces. As an application of the results and the construction we consider positive self-adjoint extensions of the modulus square operator $T^{\ast }T$ of a densely defined linear transformation T and bounded self-adjoint extensions of a symmetric operator. Krein's results on the uniqueness of positive (respectively, norm preserving) self-adjoint extensions are also revised.

中文翻译：

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正对称算子的扩展和 Krein 的唯一性准则

我们修正了 Krein 的正对称算子的可拓理论。我们通过辅助 Hilbert 空间使用因式分解的方法有几个优点：它可以应用于非密集定义的变换，并且它适用于真实空间和复杂空间。作为结果和构造的应用，我们考虑模平方算子的正自伴随扩展${吨}^{＊}吨$密集定义的线性变换T和对称算子的有界自伴随扩展。Krein 关于正（分别地，保持范数）自伴随扩展的唯一性的结果也进行了修订。