A generalisation of the form method for accretive forms and operators

TL;DR

This paper extends the form method for accretive forms to more general settings with relaxed domain conditions, characterizing when operators are m-accretive and exploring approximation and invariance properties.

Contribution

It generalizes the form method for accretive forms under relaxed domain conditions, providing characterizations and examples for m-accretive operators.

Findings

Characterization of when the associated operator is m-accretive.

Examples of degenerate phenomena in the general setting.

Results on form approximation and invariance criteria.

Abstract

The form method as popularised by Lions and Kato is a successful device to associate m-sectorial operators with suitable elliptic or sectorial forms. McIntosh generalised the form method to an accretive setting, thereby allowing to associate m-accretive operators with suitable accretive forms. Classically, the form domain is required to be densely embedded into the Hilbert space. Recently, this requirement was relaxed by Arendt and ter Elst in the setting of elliptic and sectorial forms. Here we study the prospects of a generalised form method for accretive forms to generate accretive operators. In particular, we work with the same relaxed condition on the form domain as used by Arendt and ter Elst. We give a multitude of examples for many degenerate phenomena that can occur in the most general setting. We characterise when the associated operator is m-accretive and investigate the class of operators that can be generated. For the case that the associated operator is m-accretive, we study form approximation and Ouhabaz type invariance criteria.