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m-Accretive Extensions of Friedrichs Operators

Krešimir Burazin, Marko Erceg, Sandeep Kumar Soni

TL;DR

The paper identifies $m$-accretive extensions of abstract Friedrichs operators with realisations obeying $(V)$-boundary conditions, integrating von Neumann-type decompositions via a skew-symmetric part and a positive self-adjoint part. It shows a tight correspondence between $(V)$-boundary conditions and $m$-accretive extensions, and simultaneously connects these to $(X)$- and $(M)$-boundary frameworks, providing a constructive route between $(V)$ and $(M)$ conditions. By leveraging the boundary operator $D$, the authors present explicit equivalences and decompositions that classify all $m$-accretive extensions and their $M$-operators, illustrated by concrete ODE and Dirichlet-elliptic PDE examples. The results yield a robust toolkit for understanding boundary-value problems in the abstract Friedrichs setting, with direct implications for semigroup generation and boundary-condition design in PDEs. Overall, the work unifies operator-theoretic and boundary-condition perspectives to yield a complete, constructive classification of $m$-accretive realisations.

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.

m-Accretive Extensions of Friedrichs Operators

TL;DR

The paper identifies -accretive extensions of abstract Friedrichs operators with realisations obeying -boundary conditions, integrating von Neumann-type decompositions via a skew-symmetric part and a positive self-adjoint part. It shows a tight correspondence between -boundary conditions and -accretive extensions, and simultaneously connects these to - and -boundary frameworks, providing a constructive route between and conditions. By leveraging the boundary operator , the authors present explicit equivalences and decompositions that classify all -accretive extensions and their -operators, illustrated by concrete ODE and Dirichlet-elliptic PDE examples. The results yield a robust toolkit for understanding boundary-value problems in the abstract Friedrichs setting, with direct implications for semigroup generation and boundary-condition design in PDEs. Overall, the work unifies operator-theoretic and boundary-condition perspectives to yield a complete, constructive classification of -accretive realisations.

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.
Paper Structure (8 sections, 12 theorems, 46 equations)

This paper contains 8 sections, 12 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Abstract Friedrichs operators
  3. Definition and main properties
  4. Boundary conditions for Friedrichs operators
  5. The von Neumann extension theory
  6. m-accretive extensions of abstract Friedrichs operators
  7. Interpretation of m-accretive extensions in terms of other boundary conditions
  8. On equivalence of (V) and (M) boundary conditions

Key Result

Theorem 2.3

Let a pair of linear operators $(T_0,\widetilde{T}_0)$ on $\mathcal{H}$ satisfy (T1) and (T2). Then the following holds. Assume, in addition, (T3), i.e. $(T_0,\widetilde{T}_0)$ is a joint pair of abstract Friedrichs operators. Then

Theorems & Definitions (36)

  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Definition 2.5: (V)-boundary conditions
  • Remark 2.6
  • Definition 2.7: (X)-boundary conditions
  • Definition 2.8: (M)-boundary conditions
  • Lemma 2.9
  • Theorem 2.10
  • ...and 26 more