m-Accretive Extensions of Friedrichs Operators

TL;DR

This paper characterizes all m-accretive extensions of abstract Friedrichs operators using boundary conditions, linking them to skew-symmetric components and providing unified interpretive frameworks with applications to differential equations.

Contribution

It establishes a precise correspondence between m-accretive extensions and boundary conditions, unifies boundary condition formulations, and explores their relation to skew-symmetric parts.

Findings

All m-accretive extensions correspond to (V)-boundary conditions.

Unified framework for boundary condition formulations.

Applications to differential equations demonstrate the theory.

Abstract

The introduction of abstract Friedrichs operators in 2007-an operator-theoretic framework for studying classical Friedrichs operators has led to significant developments in the field, including results on well-posedness, multiplicity, and classification. More recently, the von Neumann extension theory has been explored in this context, along with connections between abstract Friedrichs operators and skew-symmetric operators. In this work, we show that all m-accretive extensions of abstract Friedrichs operators correspond precisely to those satisfying (V)-boundary conditions. We also establish a connection between the m-accretive extensions of abstract Friedrichs operators and their skew-symmetric components. Additionally, the three equivalent formulations of boundary conditions are unified within a single interpretive framework. To conclude, we discuss a constructive relation between (V)- and (M)-boundary conditions and examine the multiplicity of the associated M-operators. We demonstrate our results on two examples, namely, the first order ordinary differential equation on an interval, with various boundary conditions, and the second-order elliptic partial differential equation with Dirichlet boundary conditions.