Some spectral properties of the streaming operator with general boundary conditions

This paper deals with the spectral study of the streaming operator with general boundary conditions defined by means of a boundary operator K. We study the positivity and the irreducibility of the generated semigroup proved in [M. Boulanouar, L’opérateur d’Advection: existence d’un C ₀-semi-groupe (I), Transp. Theory Stat. Phys. 31, 2002, 153–167], in the case ‖K‖ ⩾ 1. We also give some spectral properties of the streaming operator and we characterize the type of the generated semigroup in terms of the solution of a characteristic equation.     

Slowly changing vectors and the asymptotic finite-dimensionality of an operator semigroup

Siberian Mathematical Journal - Let X be a Banach space and let T: X → X be a linear power bounded operator. Put X 0 = {x ∈ X | T n x → 0}. We prove that if X 0 ≠ X then...     

Transfer functions of regular linear systems Part II: The system operator and the Lax-Phillips semigroup

This paper is a sequel to a paper by the second author on regular linear systems (1994), referred to here as ``Part I'. We introduce the system operator of a well-posed linear system, which for a finite-dimensional system described by , would be the -dependent matrix . In the general case, is an unbounded operator, and we show that it can be split into four blocks, as in the finite-dimensional case, but the splitting is not unique (the upper row consists of the uniquely determined blocks and , as in the finite-dimensional case, but the lower row is more problematic). For weakly regular systems (which are introduced and studied here), there exists a special splitting of where the right lower block is the feedthrough operator of the system. Using , we give representation theorems which generalize those from Part I to well-posed linear systems and also to the situation when the ``initial time' is . We also introduce the Lax-Phillips semigroup induced by a well-posed linear system, which is in fact an alternative representation of a system, used in scattering theory. Our concept of a Lax-Phillips semigroup differs in several respects from the classical one, for example, by allowing an index which determines an exponential weight in the input and output spaces. This index allows us to characterize the spectrum of and also the points where is not invertible, in terms of the spectrum of the generator of (for various values of ). The system is dissipative if and only if (with index zero) is a contraction semigroup.

     

Non-positivity of the semigroup generated by the Dirichlet-to-Neumann operator

By analysing some explicit examples we investigate the positivity and the non-positivity of the semigroup generated by the Dirichlet-to-Neumann operator associated with the operator \(\varDelta +\lambda I\) as \(\lambda \) varies. It is known that the semigroup is positive if \(\lambda <\lambda _1\) , where \(\lambda _1\) is the principal eigenvalue of \(-\varDelta \) with Dirichlet boundary conditions. We show that it is possible for the semigroup to be non-positive, eventually positive or positive and irreducible depending on \(\lambda >\lambda _1\) .     

线性正算子序列在空间C_m(D)的收敛性问题

本文将Коровкин关于线性正算子序列和线性连续多项式算子序列逼近一元连续函数的主要结果推广到ｍ维连续函数空间Ｃｍ（Ｄ）．     

Approximation of the semigroup generated by the Robin Laplacian in terms of the Gaussian semigroup

We consider the Laplacian ΔR subject to Robin boundary conditions on the space , where Ω is a smooth, bounded, open subset of RN. It is known that ΔR generates an analytic contraction semigroup. We show how this semigroup can be obtained from the Gaussian semigroup on C₀(RN) via a Trotter formula. As the main ingredient, we construct a positive, contractive, linear extension operator Eβ from to C₀(RN) which maps an operator core for ΔR into the domain of the generator of the Gaussian semigroup.     

Solutions of a class of partial differential equations with application to the Black-Scholes equation

In this paper we shall derive the solutions of a class of partial differential equations and its application to the Black-Scholes equation.     

Outwards pointing hysteresis operators and asymptotic behaviour of evolution equations

The paper deals with the long-time behaviour of evolution systems described by ODEs and PDEs with hysteresis operators. The analysis is based on two concepts. The first one is the outward pointing property of the involved hysteresis operators which implies uniform a priori bounds for solutions, the second one is related to the hysteresis modelling itself and consists in introducing a class of thermodynamically consistent generalized Prandtl–Ishlinskii operators as a model for a nonlinear elastoplastic material law. A stability result for solutions in one-dimensional visco-elasto-plasticity is derived as an illustration of the theory.     

Enhancement of the algebraic precision of a linear operator and consequences under positivity

Let Ω be a compact convex domain in and let L be a bounded linear operator that maps a subspace of C(Ω) into C(Ω). Suppose that L reproduces polynomials up to degree m. We show that for appropriately defined coefficients a_mrj the operator

On the semigroup generated by the renormalized Nelson Hamiltonian

We consider the renormalized Nelson model at a fixed total momentum P: $H_{\mathrm{ren}}(P)$; the Hamiltonian $H_{\mathrm{ren}}(P)$ is defined through an infinite energy renormalization. We prove that $e^{?\beta H_{\mathrm{ren}}(P)}$ is positivity improving for all $P\in {\mathbb{R}}^3$ and $\beta >0$ in the Fock representation.     

Analyticity of the semigroup generated by the operator of a viscous compressible fluid

The linearized initial‐boundary value problem describing small motions of the viscous, barotropic compressible fluid in a bounded vessel is studied under various boundary conditions (Dirichlet, Neumann and intermediate). It is shown that the corresponding operator generates an analytic semigroup in the space L₂. Copyright © 1999 John Wiley & Sons, Ltd.     

离散算子的本征投影与本征幂零

本文给出例子，说明离散算子本征投影一致有界推不出其本征幂零一致，有界幂零一致有界也推不出本征幂零的幂一致有界。     

On a Class of Elliptic-Parabolic Equations on Unbounded Intervals

We study a class of degenerate elliptic second order differential operators acting on some polynomial weighted function spaces on [0,+[. We show that these operators are the generators of C ₀-semigroups of positive operators which, in turn, are the transition semigroups associated with right-continuous normal Markov processes with state space [0,+]. Approximation and qualitative properties of both the semigroups and the Markov processes are investigated as well. Most of the results of the paper depend on a representation of the semigroups we give in terms of powers of particular positive operators of discrete type we introduced and studied in a previous paper.     

Generalized resolvents of linear relations generated by a nonnegative operator function and a differential elliptic-type expression

We study invertible extensions of the minimal relation generated by a nonnegative operator function and a differential elliptic-type expression. We prove that the operators inverse to such extensions are integral operators and describe such integral operators. We obtain a formula for generalized resolvents of the minimal relation.     

The oscillation of the Poisson semigroup associated to parabolic Hermite operator

Let O(P_τ~L) be the oscillation of the Possion semigroup associated with the parabolic Hermite operator L = ?_t-?+|x|~2. We show that O(P_τ~L) is bounded from L~p(R~(n+1))into itself for 1 p ∞, bounded from L~1(R~(n+1)) into weak-L~1(R~(n+1)) and bounded from L_c~∞(R~(n+1)) into BMO(R~(n+1)). In the case p = ∞ we show that the range of the image of the operator O(P_τ~L) is strictly smaller than the range of a general singular operator.