# The Non-Proper Dissipative Extensions of a Dual Pair

## TL;DR

This paper investigates the conditions under which dissipative operators of a specific form can be extended while preserving dissipativity, with applications to differential operators and explicit computational criteria.

## Contribution

It provides a necessary and sufficient condition for dissipative extensions of non-proper dissipative operators, including special cases with explicit computational forms.

## Key findings

- Derived a condition for dissipative extensions within certain domains
- Connected operator properties to dual pairs with common core
- Presented examples involving differential operators

## Abstract

We consider dissipative operators $A$ of the form $A=S+iV$, where both $S$ and $V\geq 0$ are assumed to be symmetric but neither of them needs to be (essentially) selfadjoint. After a brief discussion of the relation of the operators $S\pm iV $to dual pairs with the so called common core property, we present a necessary and suffcient condition for any extension of $A$ with domain contained in $\mathcal{D}((S-iV)^*)$ to be dissipative. We will discuss several special situations in which this condition can be expressed in a particularly nice form -- accessible to direct computations. Examples involving ordinary differential operators are given.