Continuous Semigroups Of Operators Research Articles

Generalized Hille-Phillips type functional calculus for multiparameter semigroups

For generators of n-parameter strongly continuous operator semigroups in a Banach space, we construct a Hille-Phillips type functional calculus, the symbol class of which consists of analytic functions from the image of the Laplace transform of the convolution algebra of temperate distributions supported by the positive cone ℝ+n. The image of such a calculus is described with the help of the commutant of the semigroup of shifts along the cone. The differential properties of the calculus and some examples are presented.

The characterization of a class of quantum Markov semigroups and the associated operator-valued Dirichlet forms based on Hilbert C*-module l 2(A)

In this paper, we introduce the concept of operator-valued quadratic form based on Hilbert W*-module \(l_2 \bar \otimes A\), and give a one to one correspondence between the set of positive self-adjoint regular module operators on \(l_2 \bar \otimes A\) and the set of regular quadratic forms, where A is a finite and σ-finite von Neumann algebra. Furthermore, we obtain that a strict continuous symmetric regular module operator semigroup \(\{ T_t \} _{t \in \mathbb{R}^ + } \subset L(l_2 \bar \otimes A)\) is Markovian if and only if the associated A-valued quadratic form is a Dirichlet form, where L(\(l_2 \bar \otimes A\)) is the von Neumann algebra of all adjointable module maps on \(l_2 \bar \otimes A\).

Spectral analysis of difference and differential operators in weighted spaces

This paper is concerned with describing the spectrum of the difference operator with a constant operator coefficient , which is a bounded linear operator in a Banach space . It is assumed that acts in the weighted space , , of two-sided sequences of vectors from . The main results are obtained in terms of the spectrum of the operator coefficient and properties of the weight function.Applications to the study of the spectrum of a differential operator with an unbounded operator coefficient (the generator of a strongly continuous semigroup of operators) in weighted function spaces are given.Bibliography: 23 titles.

Stability estimates for semigroups on Banach spaces

For a strongly continuous operator semigroup on a Banach space, we revisit a quantitative version of Datko's Theorem and the estimates for the constant $M$ satisfying the inequality $||T(t)|| ≤ M e^{\omega t}$, for all $t\ge0$, in terms of the norm of the convolution and other operators involved in Datko's Theorem. We use techniques recently developed by B. Helffer and J. Sjostrand for the Hilbert space case to estimate $M$ in terms of the norm of the resolvent of the generator of the semigroup in the right half-plane.

The characterization of a class of quantum Markov semigroups and the associated operator-valued Dirichlet forms based on Hilbert W*-module

In this paper, we introduce the concept of operator-valued quadratic form based on Hilbert W*-module $l_2 \bar \otimes A$ , and give a one to one correspondence between the set of positive self-adjoint regular module operators on $l_2 \bar \otimes A$ and the set of regular quadratic forms, where A is a finite and σ-finite von Neumann algebra. Furthermore, we obtain that a strict continuous symmetric regular module operator semigroup $\{ T_t \} _{t \in \mathbb{R}^ + } \subset L(l_2 \bar \otimes A)$ is Markovian if and only if the associated A-valued quadratic form is a Dirichlet form, where L( $l_2 \bar \otimes A$ ) is the von Neumann algebra of all adjointable module maps on $l_2 \bar \otimes A$ .

Analysis of a dynamic electro‐elastic problem

AbstractWe study a dynamic problem for linearly electro‐elastic materials, permitting piezoelectric effects. As a theoretical result, we state and prove the existence of a unique solution to the problem by using arguments of semi‐linear evolutionary equations and semigroups of linear continuous operators. Then we introduce a fully discrete scheme to solve the problem numerically. Under certain solution regularity assumptions, we derive an optimal order error estimate. Finally, we present some numerical results on a two‐dimensional test problem to illustrate the theoretical error estimate.

Distributional chaos for strongly continuous semigroups of operators

Distributional chaos for strongly continuous semigroups is studied and characterized. It is shown to be equivalent to the existence of a distributionally irregular vector. Finally, a sufficient condition for distributional chaos on the point spectrum of the generator of the semigroup is presented. An application to the semigroup generated in $L^2(R)$ by a translation of the Ornstein-Uhlenbeck operator is also given.

Bernstein functions and rates in mean ergodic theorems for operator semigroups

We present a functional calculus approach to the study of rates of decay in mean ergodic theorems for bounded strongly continuous operator semigroups. A central role is played by operators of the form g(A, where −A is the generator of the semigroup and g is a Bernstein function. In addition, we obtain some new results on Bernstein functions which are of independent interest.

On conditional mean ergodic semigroups of random linear operators

In this article, we prove two forms of conditional mean ergodic theorem for a strongly continuous semigroup of random isometric linear operators generated by a semigroup of measure-preserving measurable isomorphisms, one of which generalizes and improves several known important results.

Regularity and well-posedness of some quasi-linear elliptic and parabolic problems with nonlinear general Wentzell boundary conditions on nonsmooth domains

We show that on a bounded domain Ω⊂RN with Lipschitz continuous boundary ∂Ω, if 2N/(N+2)<p≤N−1, then weak solutions of the quasi-linear elliptic equation −Δpu+|u|p−2u+α1(x,u)=f in Ω with the general Wentzell boundary conditions −Δp,Γu+|∇u|p−2∂νu+|u|p−2u+α2(x,u)=g weakly on ∂Ω, are globally Hölder continuous on Ω¯. Here, Δp and Δp,Γ denote the p-Laplace operator, and the p-Laplace–Beltrami operator on the boundary, respectively. The functions α1(x,⋅) (for x∈Ω) and α2(x,⋅) (for x∈∂Ω) are continuous on R and satisfy a certain growth condition. We also obtain that a realization of the operator Δpu−α1(x,u) in C(Ω¯) with the boundary conditions [Δpu−α1(x,u)]|∂Ω−Δp,Γu+|∇u|p−2∂νu+α2(x,u)=0 on ∂Ω generates a strongly continuous semigroup of nonlinear operators on C(Ω¯).

On mean ergodic semigroups of random linear operators

In this paper, we prove a mean ergodic theorem for an almost surely bounded strongly continuous semigroup of random linear operators on a random reflexive random normed module, which generalizes and improves several known important results.

Dynamics of the heat semigroup in Jacobi analysis

Let Δ be the Jacobi Laplacian. We study the chaotic and hypercyclic behaviour of the strongly continuous semigroups of operators generated by perturbations of Δ with a multiple of the identity on Lp spaces.

On the structure of a semigroup of operators with finite-dimensional ranges

In the present paper, we describe the structure of a strongly continuous operator semigroup T(t) (where T: ℝ+ → End X and X is a complex Banach space) for which ImT(t) is a finite-dimensional space for all t > 0. It is proved that such a semigroup is always the direct sum of a zero semigroup and a semigroup acting in a finite-dimensional space. As examples of applications, we discuss differential equations containing linear relations, orbits of a special form, and the possibility of embedding an operator in a C0-semigroup.

Uniformly γ-radonifying families of operators and the stochastic Weiss conjecture

We introduce the notion of uniform -radonification of a family of operators, which unifies the notions of R-boundedness of a family of operators and -radonification of an individual operator. We study the properties of uniformly -radonifying families of operators in detail and apply our results to the stochastic abstract Cauchy problem dU(t) = AU(t)dt + B dW(t), U(0) = 0. Here, A is the generator of a strongly continuous semigroup of operators on a Banach space E, B is a bounded linear operator from a separable Hilbert space H into E, and WH is an H-cylindrical Brownian motion. When A and B are simultaneously diagonalisable, we prove that an invariant measure exists if and only if the family p R (,A )B : 2 S#} is uniformly -radonifying for some/all 0 < # < 2 , where S# is the open sector of angle # in the complex plane. This result can be viewed as a partial solution of a stochastic version of the Weiss conjecture in linear systems theory.

Exponential trichotomies and continuity of invariant manifolds

In this work, we consider the invariant manifolds for the family of equations ˙ x = Ax + f(,x), where A the is generator of a strongly continuous semigroup of linear operators in a Banach space X and f(,·) : X ! X is continuous. The existence of stable (unstable) and center-stable (center-unstable) man- ifolds for a large class of these equations has been proved in (2). We prove here that, if A admits a exponential trichotomy and f satisfies some suitable regularity hypotheses, then those manifolds are continu- ous with respect to the parameter .

Smooth Contractive Embeddings and Application to Feynman Formula for Parabolic Equations on Smooth Bounded Domains

We prove two assumptions made in an article by Butko et al. (2010) concerning the existence of a strongly continuous operator semigroup solving a Cauchy-Dirichlet problem for an elliptic differential operator in a bounded domain and the existence of a smooth contractive embedding of a core of the generator of the semigroup into the space . Based on these assumptions, a Feynman formula for the solution of the Cauchy-Dirichlet problem is constructed in the article mentioned above. In this article, we show that the assumptions are fulfilled for domains with C 4, α-smooth boundary and coefficients in C 2, α.

Transference principles for semigroups and a theorem of Peller

A general approach to transference principles for discrete and continuous operator (semi)groups is described. This allows one to recover the classical transference results of Calderón, Coifman and Weiss and of Berkson, Gillespie and Muhly and the more recent one of the author. The method is applied to derive a new transference principle for (discrete and continuous) operator semigroups that need not be groups. As an application, functional calculus estimates for bounded operators with at most polynomially growing powers are derived, leading to a new proof of classical results by Peller from 1982. The method allows for a generalization of his results away from Hilbert spaces to L p -spaces and—involving the concept of γ-boundedness—to general Banach spaces. Analogous results for strongly-continuous one-parameter (semi)groups are presented as well. Finally, an application is given to singular integrals for one-parameter semigroups.

On conditions for invertibility of difference and differential operators in weight spaces

We obtain necessary and sufficient conditions for the invertibility of the difference operator , , , whose domain is given by the condition , where , , is the Banach space of sequences (of vectors in a Banach space ) summable with weight for and bounded with respect to for , is a bounded linear operator, and is a closed -invariant subspace of . We give applications to the invertibility of differential operators with an unbounded operator coefficient (the generator of a strongly continuous operator semigroup) in weight spaces of functions.

Hypercyclic and mixing operator semigroups

AbstractWe describe a class of topological vector spaces admitting a mixing uniformly continuous operator group$\smash{\{T_t\}_{t\in\mathbb{C}^n}}$with holomorphic dependence on the parametert. This result builds on those existing in the literature. We also describe a class of topological vector spaces admitting no supercyclic strongly continuous operator semigroups$\smash{\{T_t\}_{t\geq0}}$.

Converse Lyapunov Theorems for Switched Systems in Banach and Hilbert Spaces

We consider switched systems on Banach and Hilbert spaces governed by strongly continuous one-parameter semigroups of linear evolution operators. We provide necessary and sufficient conditions for their global exponential stability, uniform with respect to the switching signal, in terms of the existence of a Lyapunov function common to all modes.