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

Abstract

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.