Everything is possible for the domain intersection ${\rm dom}\,T\cap{\rm dom}\,T*$

Arlinskiĭ, Yury; Tretter, Christiane (2020). Everything is possible for the domain intersection ${\rm dom}\,T\cap{\rm dom}\,T*$. Advances in mathematics, 374, p. 107383. Elsevier 10.1016/j.aim.2020.107383

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In this paper we show 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 domT \cap dom T^*={0}, we construct classes of operators for which dim (domT \cap domT^*)=n \in \mathbb N_0; dim (domT \cap domT^*)=\infty and at the same time codim (domT \cap domT^*)=\infty; and codim (domT \cap domT^*)=n \in \mathbb N_0; the latter includes the case that domT \cap dom T^* is dense but no core of \mathcal T and T^* and the case dom T=dom T^* for non-normal \mathcal T. We also show that all these possibilities may occur for operators \mathcal T with non-empty resolvent set such that either W(T)=\mathbb C, \mathcal T is maximal accretive but not sectorial, or \mathcal T is even maximal sectorial. Moreover, in all but one subcase \mathcal T can be chosen with compact resolvent.

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