Continuous Semigroups Of Operators Research Articles

The semi-discrete diffusion convection equation with decay

In this paper we find the fundamental solution of the semi-discrete diffusion convection equation with decay, and we show that under a certain combination of the parameters of the equation said fundamental solution constitutes a uniformly continuous semigroup of operators in the Lebesgue spaces ℓp(Z),1≤p≤∞.

On sequences of sectorial forms converging 'from above'

We present a form convergence theorem for sequences of sectorial forms and their associated semigroups in a complex Hilbert space. Roughly speaking, the approximating forms $a_n$ are all `bounded below' by the limiting form $a$, but in contrast to the previous literature there is no monotonicity hypothesis on the sequence. Moreover, the forms are not supposed to be closed or densely defined. For a sectorial form one obtains an associated linear relation, whose negative generates a degenerate strongly continuous semigroup of linear operators. Our hypotheses on the sequence of forms imply strong resolvent convergence of the associated linear relations, which in turn implies convergence of the corresponding semigroups. The result is illustrated by two examples, one of them closely related to the Galerkin method of numerical analysis.

Ces\`{a}ro-Type Operator on the Dirichlet Space of the Upper Half Plane

We construct a Ces\`{a}ro-type operator acting on Dirichlet space of the upper half plane using the approach of strongly continuous semigroups of composition operators on Banach spaces. We then determine the spectral and norm properties of the obtained Ces\`{a}ro-type operator.

Semigroups of composition operators on Hardy spaces of Dirichlet series

We consider continuous semigroups of analytic functions {Φt}t≥0 in the so-called Gordon-Hedenmalm classG, that is, the family of analytic functions Φ:C+→C+ giving rise to bounded composition operators in the Hardy space of Dirichlet series H2. We show that there is a one-to-one correspondence between continuous semigroups {Φt}t≥0 in the class G and strongly continuous semigroups of composition operators {Tt}t≥0, where Tt(f)=f∘Φt, f∈H2. We extend these results for the range p∈[1,∞). For the case p=∞, we prove that there is no non-trivial strongly continuous semigroup of composition operators in H∞. We characterize the infinitesimal generators of continuous semigroups in the class G as those Dirichlet series sending C+ into its closure. Some dynamical properties of the semigroups are obtained from a description of the Koenigs map of the semigroup.

The Exponential Attractor for Partially Dissipative Lattice System of Infinite Dimension

With the wide application of fractional differential equations, the study of dynamic systems of fractional differential equations has become one of the hotspots in the past decade. The attractor theory is one of the research directions of infinite dimensional lattice dynamical system. In paper, Eden et al. Systematically introduced the exponential attractor of deterministic autonomous dynamical system, which is a compact positive invariant set with finite fractal dimension and attracting orbits at exponential rate. The exponential attractor has a finite fractal dimension. Because of the exponential attractiveness, the exponential attractor is more stable under perturbation than the global attractor. Fitzhugh Nagumo equation is a nonlinear partial equation that describes the periodic oscillation of neuronal action potential under constant current stimulation above the threshold. For Fitzhugh Nagumo system, there are many reports on attractors, but few reports on exponential attractors. In this paper, the infinite dimensional lattice Fitzhugh Nagumo system [4] is studied. The specific contents are as follows: In part 1, the background, development and research status of FitzHugh Nagumo lattice system and the main work of this paper are introduced. In part 2, the basic concepts, inequalities and lemmas required in the research of FitzHugh Nagumo lattice system are introduced. In part 3, the initial value problem of FitzHugh Nagumo lattice system is studied. For any initial value ϕ(0), there exists a unique solution ϕ(t) of system (9) and family of mapping S(t) can generate continuous operator semigroup {S(t)} t≥0 on ,. Through lemma 3.2,3.3,3.4, it can be proved that the continuous operator semigroup {S(t)} t≥0 satisfies uniform continuity and is asymptotically compact, thus proving the existence of the exponential attractor of the system.

Research on the Reliability of a Two-Robot Security System with Early Warning Function

In this paper, the mathematical model of a kind of two-robot security system with an early warning function is studied. By using strongly continuous operator semigroup theory and Volterra integral equation theory, the properties of the semigroup of the system operator, the existence and uniqueness of nonnegative solution, and the well-posedness of solution are discussed, respectively. Under the assumption that the failure rate and repair rate of the system are constants, the equations of the robot system are transformed into an ordinary differential equation group, and then the instantaneous reliability and stable-state reliability of the system are obtained. The reliability and zero-state controllability of the system are proved. Finally, the numerical solution of the system model is obtained by using MATLAB mathematical software, and the corresponding numerical simulation diagram is given. The results show that the conclusions of numerical calculation and numerical simulation are in accordance with the results of reliability theory, thereby the reliability of the robot safety system is verified.

On abstract Cauchy problems in the frame of a generalized Caputo type derivative

In this paper, we consider a class of abstract Cauchy problems in the framework of a generalized Caputo type fractional. We discuss the existence and uniqueness of mild solutions to such a class of fractional differential equations by using properties found in the related fractional calculus, the theory of uniformly continuous semigroups of operators and the fixed point theorem. Moreover, we discuss the continuous dependence on parameters and Ulam stability of the mild solutions. At the end of this paper, we bring forth some examples to endorse the obtained results

Spectral properties of locally eventually positive operator semigroups

This paper considers strongly continuous semigroups of operators on Banach lattices which are locally eventually positive, a property that was first investigated in the context of concrete fourth-order evolution equations. We construct a simple example to show that the typical assumptions on the spectrum of the semigroup generator considered currently in the literature are far from necessary in the more general setting of local eventual positivity. Under minimal additional assumptions, we obtain results on the asymptotic behaviour of orbits, as well as necessary conditions on the peripheral point spectrum of locally eventually positive semigroups.

Weighted composition operators on the Fock space: iteration and semigroups

This paper considers discrete and continuous semigroups of (weighted) composition operators on the Fock space. For discrete semigroups consisting of powers of a single operator, the asymptotic behaviour of the semigroups is analysed. For continuous semigroups and groups, a full classification of possible semigroups is given, and the generator is calculated.

Semigroups of Composition Operators on Vector-Valued Hardy Spaces

In this article, a one-to-one correspondence between the set of all one-parameter semigroup of holomorphic self-mappings of the upper-half plane and the set of all strongly continuous one-parameter semigroup of composition operators on vector-valued Hardy space of the upper-half plane is established.

On a Certain Operator Related to Blackwell’s Markov Chain

AbstractWe present an example of a densely defined, linear operator on the $$l^{1}$$ l 1 space with the property that each basis vector of the standard Schauder basis of $$l^{1}$$ l 1 does not belong to its domain. Our example is based on the construction of a Markov chain with all states instantaneous given by D. Blackwell in 1958. In addition, it turns out that the closure of this operator is the generator of a strongly continuous semigroup of Markov operators associated with Blackwell’s chain.

The Bi-Laplacian with Wentzell Boundary Conditions on Lipschitz Domains

We investigate the Bi-Laplacian with Wentzell boundary conditions in a bounded domain Omega subseteq mathbb {R}^d with Lipschitz boundary Gamma . More precisely, using form methods, we show that the associated operator on the ground space L^2(Omega )times L^2(Gamma ) has compact resolvent and generates a holomorphic and strongly continuous real semigroup of self-adjoint operators. Furthermore, we give a full characterization of the domain in terms of Sobolev spaces, also proving Hölder regularity of solutions, allowing classical interpretation of the boundary condition. Finally, we investigate spectrum and asymptotic behavior of the semigroup, as well as eventual positivity.

A categorical approach to operator semigroups

The aim of this paper is to exploit the structure of strongly continuous operator semigroups in order to formulate a categorical framework in which a fresh perspective can be applied to past operator theoretic results. In particular, we investigate the inverse-producing Arens extension for Banach algebras (Trans. Am. Math. Soc. 88:536–548, 1958) adapted for operators and operator semigroups by Batty and Geyer (J. Oper. Theory 78(2):473–500, 2017) in this new framework, asking and answering questions using categorical language. We demonstrate that the Arens extension defines an extension functor in this setting and that it forms an adjunction with the suitably defined forgetful functor. As a by-product of this categorical framework, we also revisit the work on Banach direct sums by Lachowicz and Moszyński (Semigroup Forum 93(1):34–70, 2016). This paper can be considered as a brief exploration of the triple interface between operator semigroups, Banach algebras, and category theory.

Fractional Nonlinear Stochastic Heat Equation with Variable Thermal Conductivity

Abstract We consider a nonlinear stochastic heat equation with Riesz space-fractional derivative and variable thermal conductivity, on infinite domain. First we approximate the original problem by regularizing the Riesz space-fractional derivative. Then we prove that the approximate problem has almost surely a unique solution within a Colombeau generalized stochastic process space. In our solving procedure we use the theory of Colombeau generalized uniformly continuous semigroups of operators. At the end, we study the relation of the original and the approximate problem and prove that, under certain conditions, the derivative operators appearing in these two problems are associated. Even more, we prove that under some additional conditions, solutions of the original and the approximate problem are almost certainly associated as well (assuming that the first one almost surely exists).

Asymptotic behaviour of fast diffusions on graphs

We study a diffusion process on a finite graph with semipermeable membranes on vertices. We prove, in L^1 and L^2-type spaces that for a large class of boundary conditions, describing communication between the edges of the graph, the process is governed by a strongly continuous semigroup of operators, and we describe asymptotic behaviour of the diffusion semigroup as the diffusions’ speed increases at the same rate as the membranes’ permeability decreases. Such a process, in which communication is based on the Fick law, was studied by Bobrowski (Ann. Henri Poincaré 13(6):1501–1510, 2012) in the space of continuous functions on the graph. His results were generalized by Banasiak et al. (Semigroup Forum 93(3):427–443, 2016). We improve, in a way that cannot be obtained using a very general tool developed recently by Engel and Kramar Fijavž (Evolut. Equ. Control Theory 8(3)3:633–661, 2019), the results of J. Banasiak et al.

Direct integrals of strongly continuous operator semigroups

The goal of this article is to develop a theory for direct integrals of C0-semigroups on Hilbert spaces parallel to the recent approach by Lachowicz and Moszyński for direct sums of Banach spaces, diagonal operators, and semigroups. In it we deal with the existence and characterisation of semigroups, asymptotic rates, and questions of decomposability.

Analogs of the Lebesgue Measure and Diffusion in a Hilbert Space

We study shift-invariant measures on a real separable Hilbert space E, which are also invariant with respect to orthogonal transforms. In this article a finitely additive analogue of the Lebesgue measure is constructed. It is a non-negative finitely additive shift-invariant measure on a special ring of subsets from a space E, which is invariant with respect to orthogonal transforms. The ring contains all infinite-dimensional rectangles, whose products of side lengths are absolutely convergent. We define a Hilbert space \( \mathcal{\mathscr{H}} \), whose elements are complex-valued functions. The functions are square-integrable by some shift-invariant measure, which is also invariant with respect to rotations. We define the expected values of shift operators over random vectors, whose distribution is given by a family of Gaussian measures on a space E. The measures form a semigroup with respect to convolution. We prove that the expected values form a semigroup of contracting self-adjoint operators on a space \( \mathcal{\mathscr{H}} \). The semigroup is not strongly continuous. We also find invariant subspaces, where the semigroup is continuous in the strong operator topology. We investigate the structure of arbitrary (not necessarily continuous) operator semigroups of contracting self-adjoint transforms on a Hilbert space. We show that the suggested in Orlov et al. (Izv. Math. 80(6), 1131–1158 2016) method of the Feynman averaging is applicable to discontinuous semigroups.

Orbital and spectral aspects of hypercyclic operators and semigroups

In connection with the theory of hypercyclicity of bounded linear operators and strongly continuous semigroups of operators on Banach spaces, we make several observations about the following three topics – (i) special types of orbits, (ii) the role of large subsets of N, (iii) various spectral conditions – and thereby highlight some nuances of the theory.

$$\Gamma $$-Supercyclicity for Strongly Continuous Semigroups

We characterize the subsets $$\Gamma $$ of the complex plane $$\mathbb {C}$$ for which the notion of $$\Gamma $$ -supercyclicity for strongly continuous semigroups coincides with that of hypercyclicity. In addition, we characterize the sets $$\Gamma \subset {\mathbb {C}}$$ such that for every strongly continuous semigroup of operators $${\mathcal {T}}=(T_t)_{t\geqslant 0}$$ on a Banach space X, a vector $$x\in X$$ is hypercyclic for $${\mathcal {T}}$$ if and only if $${\mathrm {Orb}}(\Gamma x,\mathcal {T})$$ is somewhere dense in X. We derive a characterization of $$C_0$$ -semigroup versions of 1-dimensional hypercyclic subsets and Bourdon–Feldman subsets. We finally prove a multi- $$\Gamma $$ -supercyclicity result for $$C_0$$ -semigroups.

The Specification Property for $$C_0$$-Semigroups

We study one of the strongest versions of chaos for continuous dynamical systems, namely the specification property. We extend the definition of specification property for operators on a Banach space to strongly continuous one-parameter semigroups of operators, that is, $$C_0$$ -semigroups. In addition, we study the relationships of the specification property for $$C_0$$ -semigroups (SgSP) with other dynamical properties: mixing, Devaney’s chaos, distributional chaos, and frequent hypercyclicity. Concerning the applications, we provide several examples of semigroups which exhibit the SgSP with particular interest on solution semigroups to certain linear PDEs, which range from the hyperbolic heat equation to the Black–Scholes equation.