For a bounded operator $A$ in a Hilbert space $\mathcal H$ the real part of $A$ is defined by $\operatorname{Re}(A) = \frac 12(A+A^*)$. However, if $A$ is unbounded, this operator is defined on the intersection of the domains of $A$ and $A^*$. It is a symmetric operator, but it might not even be densely defined. I wonder if it's possible that this intersection is even trivial. So, here goes my
Question: Does anyone have an example of a closed and densely defined operator $A$ on an infinite-dimensional Hilbert space for which $$ \operatorname{dom}A\cap\operatorname{dom}A^* = \{0\}\,? $$
The paper "Everything is possible for the domain intersection $\operatorname{dom} T \cap \operatorname{dom} T^*$" by Yury Arlinskii and Christiane Tretter gives a construction that answers this question in the affirmative, with the additional property that $A$ can be taken $m$-sectorial (defined in the "preliminaries" section, if you have not seen it before) and with or without compact resolvent. See Theorem 3.1 of the preprint, https://arxiv.org/abs/1911.05042, and hopefully also a theorem in the published paper, https://doi.org/10.1016/j.aim.2020.107383 (behind a paywall). I do not know if this construction can be simplified if one does not care whether or not the operator $A$ is $m$-sectorial. The paper also explains how other possibilities for $\operatorname{dom} A \cap \operatorname{dom} A^*$ can be achieved.
The introduction to Arlinskii and Tretter's paper gives a good review of relevant literature from this perspective. Kato's Perturbation Theory for Linear Operators is a book with useful background information on sectorial operators. The references cited in Arlinskii and Tretter (e.g. Okazawa's reference [47] in the preprint, which is freely available on Project Euclid at http://doi.org/10.2969/jmsj/02710160) may also be of interest.