Sectorial forms and degenerate differential operators

TL;DR

This paper introduces a method to associate holomorphic semigroup generators with sectorial forms that may not be closable, broadening the applicability of semigroup theory to various differential operators.

Contribution

It develops a natural association between non-closable sectorial forms and semigroup generators, removing the need for the form to be closed or symmetric in key theorems.

Findings

Applicable to complex sectorial differential operators

Includes Dirichlet-to-Neumann operators

Handles operators with Robin or Wentzell boundary conditions

Abstract

If $a$ is a densely defined sectorial form in a Hilbert space which is possibly not closable, then we associate in a natural way a holomorphic semigroup generator with $a$. This allows us to remove in several theorems of semigroup theory the assumption that the form is closed or symmetric. Many examples are provided, ranging from complex sectorial differential operators, to Dirichlet-to-Neumann operators and operators with Robin or Wentzell boundary conditions.