$m-$accretive extensions of Friedrichs operators

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.