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

TL;DR

This paper demonstrates the diverse possible structures of the intersection of an operator's domain and its adjoint's domain for various classes of closed linear operators with non-empty resolvent set, including maximal sectorial and accretive operators.

Contribution

It constructs explicit examples showing all possible dimensions and codimensions of the domain intersection for common classes of operators, revealing the full range of possibilities.

Findings

Operators with trivial domain intersection ($oxed{0}$) exist.

Infinite-dimensional domain intersections with infinite codimension are possible.

Operators with dense but non-core domain intersections can be constructed.

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.