Some remarks on the essential spectrum of transport operators with abstract boundary conditions

1994 ◽  
Vol 35 (11) ◽  
pp. 6199-6212 ◽  
Author(s):  
Khalid Latrach
Keyword(s):  
Boundary Conditions ◽  
Essential Spectrum ◽  
Abstract Boundary

On compressions of self-adjoint extensions of a symmetric linear relation with unequal deficiency indices

2019 ◽  
Vol 16 (4) ◽  
pp. 567-587
Author(s):  
Vadim Mogilevskii
Keyword(s):  
Hilbert Space ◽  
Boundary Conditions ◽  
Linear Relation ◽  
Deficiency Indices ◽  
Space Extension ◽  
Limit Value ◽  
Abstract Boundary ◽  
Symmetric Linear Relation ◽  
Symmetric Operators

Let $A$ be a symmetric linear relation in the Hilbert space $\gH$ with unequal deficiency indices $n_-A <n_+(A)$. A self-adjoint linear relation $\wt A\supset A$ in some Hilbert space $\wt\gH\supset \gH$ is called an (exit space) extension of $A$. We study the compressions $C (\wt A)=P_\gH\wt A\up\gH$ of extensions $\wt A=\wt A^*$. Our main result is a description of compressions $C (\wt A)$ by means of abstract boundary conditions, which are given in terms of a limit value of the Nevanlinna parameter $\tau(\l)$ from the Krein formula for generalized resolvents. We describe also all extensions $\wt A=\wt A^*$ of $A$ with the maximal symmetric compression $C (\wt A)$ and all extensions $\wt A=\wt A^*$ of the second kind in the sense of M.A. Naimark. These results generalize the recent results by A. Dijksma, H. Langer and the author obtained for symmetric operators $A$ with equal deficiency indices $n_+(A)=n_-(A)$.

Fixed point theorems for block operator matrix and an application to a structured problem under boundary conditions of Rotenberg’s model type

2014 ◽  
Vol 64 (1) ◽  
Author(s):  
Afif Amar ◽  
Aref Jeribi ◽  
Bilel Krichen
Keyword(s):  
Boundary Conditions ◽  
Fixed Point Theorems ◽  
Existence Of Solutions ◽  
Coupled System ◽  
Operator Matrix ◽  
Model Type ◽  
Block Operator Matrix ◽  
Growing Cell ◽  
Block Operator ◽  
Abstract Boundary

AbstractIn this manuscript, we introduce and study the existence of solutions for a coupled system of differential equations under abstract boundary conditions of Rotenberg’s model type, this last arises in growing cell populations. The entries of block operator matrix associated to this system are nonlinear and act on the Banach space X p:= L p([0, 1] × [a, b]; dµ dv), where 0 ≤ a < b < ∞; 1 < p < ∞.

Differential Operators with Abstract Boundary Conditions

1978 ◽  
Vol 30 (02) ◽  
pp. 262-288 ◽  
Author(s):  
R. C. Brown
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Vector Space ◽  
Differential Operators ◽  
Dimensional Vector ◽  
Absolutely Continuous ◽  
Abstract Boundary ◽  
Image Position ◽  
Vector Valued Functions ◽  
Vector Valued

Suppose F is a topological vector space. Let ACm ≡ ACm[a, b] be the absolutely continuous m-dimensional vector valued functions y on the compact interval [a, b] with essentially bounded components. Consider the boundary value problem where A0, A are respectively... operator with range in F.

scholarly journals Stability of some essential B-spectra of pencil operators and application

2021 ◽  
Vol 36 (1) ◽  
pp. 63-80
Author(s):  
A. Ben Ali ◽  
M. Boudhief ◽  
N. Moalla
Keyword(s):  
Banach Space ◽  
Differential Operator ◽  
Linear Operator ◽  
Boundary Conditions ◽  
Operator Pencil ◽  
Abstract Boundary

In this paper, we give some results on the essential B-spectra of a linear operator pencil, which are used to determine the essential B-spectra of an integro-differential operator with abstract boundary conditions in the Banach space Lp([−a, a] × [−1, 1]), p ≥ 1 and a > 0.

COMPACTNESS RESULTS FOR TRANSPORT EQUATIONS AND APPLICATIONS

2001 ◽  
Vol 11 (07) ◽  
pp. 1181-1202 ◽  
Author(s):  
KHALID LATRACH
Keyword(s):  
Boundary Conditions ◽  
Collision Operator ◽  
Transport Equations ◽  
Boundary Operator ◽  
Systematic Analysis ◽  
Boundary Operators ◽  
Variable Space ◽  
Abstract Boundary ◽  
Two Parameters ◽  
Transport Problems

The goal of this paper is to give a systematic analysis of compactness properties for transport equations with general boundary conditions where an abstract boundary operator relates the incoming and outgoing fluxes. The analysis involves two parameters: The velocity measure and the collision operator. Hence, for a large class of (velocity) measures and under appropriate assumptions on scattering operators compactness results are obtained. Using the positivity (in the lattice sense) and the comparison arguments by Dodds–Fremlin, their converses are derived, and necessary conditions for some remainder term of the Dyson–Phillips expansion to be compact are given. Our results are independent of the properties of the boundary operators and play a crucial role in the understanding of the time asymptotic structure of evolution transport problems. Also, although solutions of transport equations propagate singularities (due to the hyperbolic nature of the operator), they bring the regularity in the variable space (regardless of the boundary operator). We end the paper by applying the obtained results to discuss the existence of solutions to a nonlinear boundary value problem and to describe in detail the various essential spectra of transport operators with abstract boundary conditions.

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