arXiv subjects

Sandeep Kumar Soni

Publications and source records attributed to Sandeep Kumar Soni.

5 recordsLinked to original sources

arXiv2026

Boundary quadruples and bijective realisations of abstract Friedrichs operators

The theory of boundary quadruples and boundary triples is well-studied for symmetric and skew-symmetric operators and in general for dual-pairs. This paper adapts a suitable version for abstract Friedrichs operators and addresses the following questions: which parameters yield bijective realisations, and which parameters yield $m-$accretive realisations. We study a boundary-quadruple framework in which closed realisations are parametrised by closed relations in a boundary space. This yields the intrinsic criterion \[T_\Theta\ \rm{is} \ \rm{bijective}\iff \lK=\Theta\dotplus \Gamma(\ker T_1) \;.\] For bounded operator parameters $\phi:\lK_1\to \lK_0$ in the boundary space, we introduce the reference operator \[Q_0=\Gamma_1(\Gamma_0|_{\ker T_1})^{-1}\,,\] prove that $\|Q_0\|< 1$, and obtain the exact criterion \[T_\phi \ \rm{is} \ \rm{bijective} \iff \I_{\lK_0}-\phi Q_0\ \rm{is} \ \rm{bijective}\;.\] Consequently, every non-expansive parameter gives a bijective realisation with signed boundary map, which is also $m-$accretive. An existence criterion for boundary quadruples and boundary triples is established in terms of (V)-boundary conditions. The multiplicity of $M-$operators associated with a fixed (V)-boundary condition is addressed in an explicit way and a parametrisation of such operators is given. The theory is illustrated by a first-order ordinary differential operator and by the stationary diffusion equation, where $Q_0$ is identified as a Cayley transform of the Dirichlet-to-Neumann operator.

math.FA↗

arXiv2025

m-Accretive Extensions of Friedrichs Operators

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.

math.AP↗

arXiv2024

Friedrichs systems on an interval

There has been significant developments in the classification of boundary conditions of positive symmetric systems, also known as Friedrichs systems, after the introduction of operator theoretic framework. We take a step forward towards applying the abstract theory to the classical framework by studying Friedrichs systems on an interval. Dealing with some difficulties related to the smoothness of eigenvectors, here we present an explicit expression for the dimensions of the kernels of Friedrichs operators only in terms of the values of the coefficients at the end-points of the interval. In particular, this allows for a characterisation of all admissible boundary conditions, i.e.~those leading to bijective realisations.

math.AP↗

arXiv2023

The von Neumann extension theory for abstract Friedrichs operators

The theory of abstract Friedrichs operators was introduced some fifteen years ago with the aim of providing a more comprehensive framework for the study of positive symmetric systems of first-order partial differential equations, nowadays better known as (classical) Friedrichs systems. Since then, the theory has not only been frequently applied in numerical and analytical research of Friedrichs systems, but has continued to evolve as well. In this paper, we provide an explicit characterisation and a classification of abstract Friedrichs operators. More precisely, we show that every abstract Friedrichs operator can be written as the sum of a skew-symmetric operator and a bounded self-adjoint strictly positive operator. Furthermore, we develop a classification of realisations of abstract Friedrichs operators in the spirit of the von Neumann extension theory, which, when applied to the symmetric case, extends the classical theory.

math.AP↗

arXiv2022

Classification of classical Friedrichs differential operators: One-dimensional scalar case

The theory of abstract Friedrichs operators, introduced by Ern, Guermond and Caplain (2007), proved to be a successful setting for studying positive symmetric systems of first order partial differential equations (Friedrichs, 1958), nowadays better known as Friedrichs systems. Recently, Antonić, Michelangeli and Erceg (2017) presented a purely operator-theoretic description of abstract Friedrichs operators, allowing for application of the universal operator extension theory (Grubb, 1968). In this paper we make a further theoretical step by developing a decomposition of the graph space (maximal domain) as a direct sum of the minimal domain and the kernels of corresponding adjoints. We then study one-dimensional scalar (classical) Friedrichs operators with variable coefficients and present a complete classification of admissible boundary conditions.

math.AP↗