Extensions of dissipative operators with closable imaginary part

TL;DR

This paper characterizes when extensions of dissipative operators preserve dissipativity, providing conditions for maximal accretiveness in specific classes of operators, with applications to symmetric and singular operators.

Contribution

It offers a necessary and sufficient condition for extending dissipative operators while maintaining their dissipative property, expanding understanding of operator extensions.

Findings

Characterization of extensions preserving dissipativity.

Description of maximally accretive extensions for positive symmetric operators.

Analysis of maximally dissipative extensions for singular first-order operators.

Abstract

Given a dissipative operator $A$ on a complex Hilbert space $\mathcal{H}$ such that the quadratic form $f\mapsto \mbox{Im}\langle f,Af\rangle$ is closable, we give a necessary and sufficient condition for an extension of $A$ to still be dissipative. As applications, we describe all maximally accretive extensions of strictly positive symmetric operators and all maximally dissipative extensions of a highly singular first-order operator on the interval.