Everything is possible for the domain intersection dom T \cap dom T*

This paper shows that for the domain intersection $\text{dom}T\cap \text{dom}{T}^{\ast }$ 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 $\dom T\cap\dom T^={0}$,weconstructclassesofoperatorsforwhich$\dim(\dom T\cap\dom T^)= n \in \dN_0$;$\dim(\dom T\cap\dom T^)= \infty$andatthesametime$\codim(\dom T\cap\dom T^)=\infty$;and$\codim(\dom T\cap\dom T^)= n \in \dN_0$;thelatterincludesthecasethat$\dom T\cap\dom T^$ is dense but no core of $T$ and $T^{\ast }$ and the case $\text{dom}T=\text{dom}{T}^{\ast }$ for non-normal $T$. We also show that all these possibilities may occur for operators $T$ with non-empty resolvent set such that either $W(T)=\text{dC}$, $T$ 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.