On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-Poisson Semigroup Associated with Elliptic Systems

We study the infinitesimal generator of the Poisson semigroup in L ^p associated with homogeneous, second-order, strongly elliptic systems with constant complex coefficients in the upper-half space, which is proved to be the Dirichlet-to-Normal mapping in this setting. Also, its domain is identified as the linear subspace of the L ^p -based Sobolev space of order one on the boundary of the upper-half space consisting of functions for which the Regularity problem is solvable. Moreover, for a class of systems containing the Lamé system, as well as all second-order, scalar elliptic operators, with constant complex coefficients, the action of the infinitesimal generator is explicitly described in terms of singular integral operators whose kernels involve first-order derivatives of the canonical fundamental solution of the given system. Furthermore, arbitrary powers of the infinitesimal generator of the said Poisson semigroup are also described in terms of higher order Sobolev spaces and a higher order Regularity problem for the system in question. Finally, we indicate how our techniques may be adapted to treat the case of higher order systems in graph Lipschitz domains.     

Solutions of the BBGKY hierarchy for a system of hard spheres with inelastic collisions

The problem of the existence of solutions of the hierarchy for the sequence of correlation functions is investigated in the direct sum of spaces of summable functions. We prove the existence and uniqueness of solutions, which are represented through a semigroup of bounded strongly continuous operators. The infinitesimal generator of the semigroup coincides on a certain everywhere dense set with the operator on the right-hand side of the hierarchy. For initial data from this set, solutions are strong; for general initial data, they are generalized ones. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 3, pp. 371–380, March, 2006.     

Strongly continuous semigroups of operators generated by systems of pseudodifferential operators in weighted <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

This paper considers semigroups of operators generated by pseudodifferential operators in weighted L _p -spaces of vector functions on \mathbbRⁿ {\mathbb{R}^n} (or on a compact manifold without boundary). Sufficient conditions for a semigroup to be strongly continuous and analytic are obtained, conditions for it to be completely continuous are found, and the distribution of the eigenvalues of its infinitesimal generator is examined. Also, an integral representation that singles out the principal term of the semigroup as t → 0+ is established.     

Semigroup Properties and the Crandall Liggett Approximation for a Class of Differential Equations with State-Dependent Delays

We present an approach for the resolution of a class of differential equations with state-dependent delays by the theory of strongly continuous nonlinear semigroups. We show that this class determines a strongly continuous semigroup in a closed subset of C^0, 1. We characterize the infinitesimal generator of this semigroup through its domain. Finally, an approximation of the Crandall-Liggett type for the semigroup is obtained in a dense subset of (C, ‖·‖_∞). As far as we know this approach is new in the context of state-dependent delay equations while it is classical in the case of constant delay differential equations.     

Some <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> properties of semigroups of measures on Lie groups

We investigate the induced action of convolution semigroups of probability measures on Lie groups on the L ²-space of Haar measure. Necessary and sufficient conditions are given for the infinitesimal generator to be self-adjoint and the associated symmetric Dirichlet form is constructed. We show that the generated Markov semigroup is trace-class if and only if the measures have a square-integrable density. Two examples are studied in some depth where the spectrum can be explicitly computed, these being the n-torus and Riemannian symmetric pairs of compact type.     

Invariant set criteria for CO semigroups

The purpose of this note is to describe conditions that guarantee the invariance of convex sets for strongly continuous semigroups of linear operators. The criteria is expressed in terms of subtangential properties of the semigroup and its infinitesimal generator. These ideas include various recent results of a similar nature.     

On the Relation between Stability of Continuous- and Discrete-Time Evolution Equations via the Cayley Transform

In this paper we investigate and compare the properties of the semigroup generated by A, and the sequence where A_d = (I + A) (I − A)⁻¹. We show that if A and A⁻¹ generate a uniformly bounded, strongly continuous semigroup on a Hilbert space, then A_d is power bounded. For analytic semigroups we can prove stronger results. If A is the infinitesimal generator of an analytic semigroup, then power boundedness of A_d is equivalent to the uniform boundedness of the semigroup generated by A.     

Robust controller for systems with exponentially stable strongly continuous semigroups

The theory of robust controllers is extended to the case where we have boundary and/or distributed control and the system operator is an infinitesimal generator of a strongly continuous exponentially stable semigroup. An example on hyperbolic systems is presented.     

Robustness of strongly and polynomially stable semigroups

In this paper we study the robustness properties of strong and polynomial stability of semigroups of operators. We show that polynomial stability of a semigroup is robust with respect to a large and easily identifiable class of perturbations to its infinitesimal generator. The presented results apply to general polynomially stable semigroups and bounded perturbations. The conditions on the perturbations generalize well-known criteria for the preservation of exponential stability of semigroups. We also show that the general results can be improved if the perturbation is of finite rank or if the semigroup is generated by a Riesz-spectral operator. The theory is applied to deriving concrete conditions for the preservation of stability of a strongly stabilized one-dimensional wave equation.     

Generation of Analytic Semigroups by Elliptic Operators with Dirichlet Boundary Conditions in a Cylindrical Domain

We consider an elliptic operator of second order with Dirichlet boundary conditions in a cylindrical domain. We show that a suitable interpretation of this operator in a certain space of continuous functions vanishing on the boundary is the infinitesimal generator of an analytic semigroup in the space of continuous functions. We prove several inclusions of the domain of the infinitesimal generator and of the real interpolation spaces between this domain and the basic space of continuous functions.     

一个双曲-椭圆耦合系统解的存在唯一性

本文研究了一个双曲-椭圆耦合系统.通过能量方法建立了有关微分算子的一些先验估计,构造了一个闭线性算子,证明了该闭线性算子为一个有界收缩线性算子半群的无穷小生成元.在此基础上,利用半群理论具体证明了双曲-椭圆耦合系统解的存在唯一性.     

Approximate solutions of abstract differential equations

The methods of arbitrarily high orders of accuracy for the solution of an abstract ordinary differential equation are studied. The right-hand side of the differential equation under investigation contains an unbounded operator which is an infinitesimal generator of a strongly continuous semigroup of operators. Necessary and sufficient conditions are found for a rational function to approximate the given semigroup with high accuracy. The research was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503.     

Robustness of strong stability of semigroups

In this paper we study the preservation of strong stability of strongly continuous semigroups on Hilbert spaces. In particular, we study a situation where the generator of the semigroup has a finite number of spectral points on the imaginary axis and the norm of its resolvent operator is polynomially bounded near these points. We characterize classes of perturbations preserving the strong stability of the semigroup. In addition, we improve recent results on preservation of polynomial stability of a semigroup under perturbations of its generator. Theoretic results are illustrated with an example where we consider the preservation of the strong stability of a multiplication semigroup.     

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Under the condition that the coefficients are Lipschitz continuous, we study the infinitesimal generator of Markov semigroup corresponding to the multivalued stochastic equation. In order to provide a core of the infinitesimal generator, we investigate the associated multivalued elliptic equation and its viscosity solutions.     

On the stability of differentiability of semigroups

LetA be the infinitesimal generator of aC ₀-semigroup. The semigroup generated byA is called differentiable ifA exp (At) is bounded for everyt>0. In this note, an example is given of an operatorA and a bounded operatorB such that the semigroup generated byA is differentiable but the semigroup generated byA+B is not. This gives a negative answer to a question of Pazy.     

线性算子双半群的扰动定理

本文研究有界线性算子强连续双半群的扰动问题。文中首先研究与强连续双半群母元有关的算子方程的可解性与算子的相似性。在此基础上证明了在一定条件下可化为指数衰减的强连续双半群经适当扰动后仍是一个可化为指数衰减的强连续双半群。     

Quantum random walks and their convergence to Evans-Hudson flows

Using coordinate-free basic operators on toy Fock spaces, quantum random walks are defined following the ideas of Attal and Pautrat. Extending the result for one dimensional noise, strong convergence of quantum random walks associated with bounded structure maps to Evans-Hudson flow is proved under suitable assumptions. Starting from the bounded generator of a given uniformly continuous quantum dynamical semigroup on a von Neumann algebra, we have constructed quantum random walks which converges strongly and the strong limit gives an Evans-Hudson dilation for the semigroup.     

Long time behavior of heat kernels of operators with unbounded drift terms

Given a second-order elliptic operator on Rd, with bounded diffusion coefficients and unbounded drift, which is the generator of a strongly continuous semigroup on L²(Rd) represented by an integral, we study the time behavior of the integral kernel and prove estimates on its decay at infinity. If the diffusion coefficients are symmetric, a local lower estimate is also proved.     

On the stabilizability problem in Banach space

The model is a linear system defined on Banach (state and control) spaces, with the operator acting on the state only the infinitesimal generator of a strongly continuous semigroup. The stabilizability problem of expressing the control through a bounded operator acting on the state as to make the resulting feedback system globally asymptotically stable is considered. On the negative side, and in contrast with the finite dimensional theory, a few counter examples are given of systems which are densely controllable in the space and yet are not stabilizable, even if some further “nice properties” hold. Use is made of the notion of essential spectrum and its stability under relatively compact perturbations. On the positive side, it is shown, however, that for large classes of systems of physical interest (classical selfadjoint boundary value problems, delay equations, etc.) controllability on a suitable finite dimensional subspace still yields stabilizability on the whole space.     

Periodic strongly continuous semigroups

Summary It is shown that three possible definitions of periodicity of a strongly continuous semigroup are equivalent. A complete characterization of periodicity is obtained in terms of the infinitesimal generator. An example is given to illustrate how the results tie in with the classical theory of periodic continuous functions. Finally some results are obtained for the case when the underlying space is a Hilbert space. Entrata in Redazione il 20 ottobre 1976. The research for this paper was done while the author was supported by the Netherlands Organization for the Advancement of Pure Research (Z.W.O.).