# 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

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.

#### Item Type:

Journal Article (Original Article)

#### Division/Institute:

08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematics

#### UniBE Contributor:

Tretter, Christiane

#### Subjects:

500 Science > 510 Mathematics

0001-8708

Elsevier

#### Funders:

[4] Swiss National Science Foundation

English

Sebastiano Don

#### Date Deposited:

28 Jan 2021 18:38

31 Jan 2021 02:59

#### Publisher DOI:

10.1016/j.aim.2020.107383

1911.05042

#### Uncontrolled Keywords:

Accretive operator, Sectorial operator, Numerical range, Domain intersection

10.48350/151234

#### URI:

https://boris.unibe.ch/id/eprint/151234