论文标题
无界的操作员具有自我伴侣或正常功能以及一些相关的结果
Unbounded operators having self-adjoint or normal powers and some related results
论文作者
论文摘要
我们表明，一个密集定义的可关闭运算符$ a $，以便不一定要关闭$ a^2 $的分解集。然后将此结果扩展到多项式$ p（a）$的情况。我们还通过Sebestyén-Tarcsay概括了J. von Neumann结果的相反。还给出了其他有趣的后果，其中之一是证明，如果$ t $是准（无界）操作员，那么$ t^n $对于某些$ n \ geq2 $是正常的，那么$ t $是正常的。到Pietrzycki-Stochel的最新结果，我们推断出一个封闭的亚正态算子，因此$ t^n $是正常的，必须是正常的。 另一个显着的结果是，不正常的操作员$ a $ a $是否有界，因此$ a^p $和$ a^q $是某些共阵数$ p $和$ q $的自我偶像，这是自我争夺。还表明，可逆运算符（是否有限制）$ a $ $ a^p $和$ a^q $对于某些联合总数$ p $和$ q $是正常的。使用Bézout的定理在算术中显示了这两个结果。
We show that a densely defined closable operator $A$ such that the resolvent set of $A^2$ is not empty is necessarily closed. This result is then extended to the case of a polynomial $p(A)$. We also generalize a recent result by Sebestyén-Tarcsay concerning the converse of a result by J. von Neumann. Other interesting consequences are also given, one of them being a proof that if $T$ is a quasinormal (unbounded) operator such that $T^n$ is normal for some $n\geq2$, then $T$ is normal. By a recent result by Pietrzycki-Stochel, we infer that a closed subnormal operator such that $T^n$ is normal, must be normal. Another remarkable result is the fact that a hyponormal operator $A$, bounded or not, such that $A^p$ and $A^q$ are self-adjoint for some co-prime numbers $p$ and $q$, is self-adjoint. It is also shown that an invertible operator (bounded or not) $A$ for which $A^p$ and $A^q$ are normal for some co-prime numbers $p$ and $q$, is normal. These two results are shown using Bézout's theorem in arithmetic.
