This paper shows that for the domain intersection $\dom T\cap\dom T^*$ of a closed linear operator and its Hilbert space adjoint everything is possible for very common classes of operators with non-empty resolvent set. Apart from the most striking case of a maximal sectorial operator with $\dom T\cap\dom T^*=\{0\}$, we construct classes of operators for which $\dim(\dom T\cap\dom T^*)= n \in \dN_0$; $\dim(\dom T\cap\dom T^*)= \infty$ and at the same time $\codim(\dom T\cap\dom T^*)=\infty$; and $\codim(\dom T\cap\dom T^*)= n \in \dN_0$; the latter includes~the case that $\dom T\cap\dom T^*$ is dense but no core of $T$ and $T^*$ and the case $\dom T=\dom T^*$ for non-normal $T$. We also show that all these possibilities may occur for operators $T$ with non-empty resolvent set such that either $W(T)=\dC$, $T$ is maximal accretive but not sectorial, or $T$ is even maximal sectorial. Moreover, in all but one subcase $T$ can be chosen with compact resolvent.

中文翻译：

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对于域交集 dom T ∩ dom T⁎ 一切皆有可能

本文表明，对于封闭线性算子的域交集 $\dom T\cap\dom T^*$ 及其 Hilbert 空间伴随，对于具有非空解析集的非常常见的算子类，一切都是可能的。除了具有 $\dom T\cap\dom T^*=\{0\}$ 的最大扇形算子的最引人注目的情况外，我们构造了 $\dim(\dom T\cap\dom T ^*)= n \in \dN_0$; $\dim(\dom T\cap\dom T^*)= \infty$ 并且同时 $\codim(\dom T\cap\dom T^*)=\infty$; 和 $\codim(\dom T\cap\dom T^*)= n \in \dN_0$; 后者包括~$\dom T\cap\dom T^*$密集但没有$T$和$T^*$核心的情况以及$\dom T=\dom T^*$为非核心的情况-正常$T$。我们还表明，所有这些可能性都可能发生在具有非空解析集的运算符 $T$ 中，使得 $W(T)=\dC$、$T$ 是最大增值但不是扇区，或 $T$ 甚至是最大的部门。此外，除了一个子案例 $T$ 之外的所有子案例都可以用紧凑的解析器来选择。