无界算子：（平方）根，零值，可闭性及相关可逆性结果
Unbounded operators: (square) roots, nilpotence, closability and some related invertibility results
摘要：
本文主要研究线性算子的任意无界平方根及其一些基本性质。本文包含了许多例子和反例。作为例证，我们给出了恒等算子和零算子的显式处处定义的无界不可闭$nth$根。我们还证明了一个无任何不可闭平方根的不可闭无界算子。在其他后果中，我们有一种方法找到处处定义的双射算子，处处定义的算子是满射而不是内射，处处定义的算子是内射而不是满射。给出了关于零的一些相关结果。
英文摘要：
In this paper, we are mainly concerned with studying arbitrary unbounded square roots of linear operators as well as some of their basic properties. The paper contains many examples and counterexamples. As an illustration, we give explicit everywhere defined unbounded non-closable $nth$ roots of the identity operator as well as the zero operator. We also show a non-closable unbounded operator without any non-closable square root. Among other consequences, we have a way of finding everywhere defined bijective operators, everywhere defined operators which are surjective without being injective and everywhere defined operators which are injective without being surjective. Some related results on nilpotence are also given.
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