Continuous One-parameter Semigroups Research Articles

Controllability and observability of conformable fractional finite dimensional linear systems

In this research paper, we investigate the controllability and observability of continuous-time fractional dynamical systems. The characterisation of all fractional continuous one-parameter semi groups on a finite dimensional vector space as fractional matrix-valued exponential functions is given. This characterisation is used to express the solution of continuous-time fractional dynamical system. Furthermore, we analyse controllability and observability of linear conformable fractional systems in the mathematical context of linear operators on a finite dimensional vector space. By defining the controllability and observability maps, we establish necessary and sufficient conditions for controllability and observability of this type of systems. We also present the Hautus test for controllability and observability, and introduce controllability and observability Gramians, both for finite and infinite time interval. Finally, we present several illustrative applications and examples to validate all the obtained results.

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.

Strongly elliptic operators and exponentiation of operator Lie algebras

An intriguing feature which is often present in theorems regardingthe exponentiation of Lie algebras of unbounded linear operators onBanach spaces is the assumption of hypotheses on the Laplacianoperator associated with a basis of the operator Lie algebra.The main objective of this work is to show that one can substitutethe Laplacian by an arbitrary operator in the enveloping algebra andstill obtain exponentiation, as long as its closure generates astrongly continuous one-parameter semigroup satisfying certain normestimates, which are typical in the theory of strongly ellipticoperators.

One-parameter groups of orthogonality preservers on C$$^*$$-algebras

We establish a more precise description of those surjective or bijective continuous linear operators preserving orthogonality between C\(^*\)-algebras. The new description is applied to determine all uniformly continuous one-parameter semigroups of orthogonality preserving operators on an arbitrary C\(^*\)-algebra. We prove that given a family \(\{T_t: t\in {\mathbb {R}}_0^{+}\}\) of orthogonality preserving bounded linear bijections on a general C\(^*\)-algebra A with \(T_0=Id\), if for each \(t\ge 0,\) we set \(h_t = T_t^{**} (1)\) and we write \(r_t\) for the range partial isometry of \(h_t\) in \(A^{**},\) and \(S_t\) stands for the triple isomorphism on A associated with \(T_t\) satisfying \(h_t^* S_t(x)\) \(= S_t(x^*)^* h_t\), \(h_t S_t(x^*)^* =\) \( S_t(x) h_t^*\), \(h_t r_t^* S_t(x) =\) \(S_t(x) r_t^* h_t\), and \(T_t(x) = h_t r_t^* S_t(x) = S_t(x) r_t^* h_t, \hbox { for all } x\in A,\) the following statements are equivalent: (a) \(\{T_t: t\in {\mathbb {R}}_0^{+}\}\) is a uniformly continuous one-parameter semigroup of orthogonality preserving operators on A; (b) \(\{S_t: t\in {\mathbb {R}}_0^{+}\}\) is a uniformly continuous one-parameter semigroup of surjective linear isometries (i.e., triple isomorphisms) on A (and hence there exists a triple derivation \(\delta \) on A such that \(S_t = e^{t \delta }\) for all \(t\in {\mathbb {R}}\)), the mapping \(t\mapsto h_t \) is continuous at zero, and the identity \( h_{t+s} = h_t r_t^* S_t^{**} (h_s),\) holds for all \(s,t\in {\mathbb {R}}.\)

Roots of completely positive maps

We introduce the concept of completely positive roots of completely positive maps on operator algebras. We do this in different forms: as asymptotic roots, proper discrete roots and as continuous one-parameter semigroups of roots. We present structural and general existence and non-existence results, some special examples in settings where we understand the situation better, and several challenging open problems. Our study is closely related to Elfving's embedding problem in classical probability and the divisibility problem of quantum channels.

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.

Subordination principle for space-time fractional evolution equations and some applications

ABSTRACTThe abstract Cauchy problem for the fractional evolution equation with the Caputo derivative of order and operator , , is considered, where generates a strongly continuous one-parameter semigroup on a Banach space. Subordination formulae for the solution operator are derived, which are integral representations containing a subordination kernel (a scalar probability density function) and a -semigroup of operators. Some properties of the subordination kernel are established and representations in terms of Mainardi function and Lévy extremal stable densities are derived. Applications of the subordination formulae are given with a special focus on the multi-dimensional space-time fractional diffusion equation.

Quantum spin chain dissipative mean-field dynamics

We study the emergent dynamics resulting from the infinite volume limit of the mean-field dissipative dynamics of quantum spin chains with clustering, but not time-invariant states. We focus upon three algebras of spin operators: the commutative algebra of mean-field operators, the quasi-local algebra of microscopic, local operators and the collective algebra of fluctuation operators. In the infinite volume limit, mean-field operators behave as time-dependent, commuting scalar macroscopic averages while quasi-local operators, despite the dissipative underlying dynamics, evolve unitarily in a typical non-Markovian fashion. Instead, the algebra of collective fluctuations, which is of bosonic type with time-dependent canonical commutation relations, undergoes a time-evolution that retains the dissipative character of the underlying microscopic dynamics and exhibits non-linear features. These latter disappear by extending the time-evolution to a larger algebra where it is represented by a continuous one-parameter semigroup of completely positive maps. The corresponding generator is not of Lindblad form and displays mixed quantum-classical features, thus indicating that peculiar hybrid systems may naturally emerge at the level of quantum fluctuations in many-body quantum systems endowed with non time-invariant states.

Symmetries Versus Conservation Laws in Dynamical Quantum Systems: A Unifying Approach Through Propagation of Fixed Points

We unify recent Noether-type theorems on the equivalence of symmetries with conservation laws for dynamical systems of Markov processes, of quantum operations, and of quantum stochastic maps, by means of some abstract results on propagation of fixed points for completely positive maps on $$C^*$$ -algebras. We extend most of the existing results with characterisations in terms of dual infinitesimal generators of the corresponding strongly continuous one-parameter semigroups. By means of an ergodic theorem for dynamical systems of completely positive maps on von Neumann algebras, we show the consistency of the condition on the standard deviation for dynamical systems of quantum operations, and hence of quantum stochastic maps as well, in case the underlying Hilbert space is infinite dimensional.

On the structure of C0-semigroups of holomorphic Carathéodory isometries in Hilbert space

We establish closed formulas for all strongly continuous one-parameter semigroups of holomorphic Carathéodory isometries of the unit ball of a Hilbert space in terms of spectral resolutions of skew self-adjoint dilations related to the Reich–Shoikhet nonlinear infinitesimal generator.

Completely monotone functions and some classes of fractional evolution equations

The abstract Cauchy problem for the distributed order fractional evolution equation in the Caputo and in the Riemann–Liouville sense is studied for operators generating a strongly continuous one-parameter semigroup on a Banach space. Continuous as well as discrete distribution of fractional time-derivatives of order less than one is considered. The problem is reformulated as an abstract Volterra integral equation. It is proven that its kernel satisfies certain complete monotonicity properties. Based on these properties, the well-posedness of the problem is established and a series expansion of the solution is obtained. In case of ordered Banach space this representation implies positivity of the solution operator. In addition, a subordination formula is obtained.

Mixing type theorems for one-parameter semigroups of operators

In this paper, we present some results concerning strong and weak mixing properties of continuous one-parameter semigroups of operators on Banach spaces. We study also some stability problems for continuous one-parameter semigroups on Hilbert spaces. Some related problems are also discussed.

E0-semigroups of type II0 and q-purity: Boundary weight maps of range rank one and two

A CP-flow over a separable Hilbert space K is a continuous one-parameter semigroup of completely positive maps on B(K⊗L2(0,∞)) which is intertwined by the right shift semigroup, and CP-flows are obtained from boundary weight maps. In this paper we generalize concepts of q-purity considered previously, by defining an E0-semigroup to be q-pure if it is a CP-flow and its set of CP-flow subordinates is totally ordered by subordination. We provide a complete description of all q-pure E0-semigroups of type II0 arising from boundary weight maps with range rank one, and we provide a criterion to determine if two such boundary weight maps give rise to cocycle conjugate q-pure E0-semigroups when dimK<∞. We also show that boundary weight maps of range rank two do not give rise to q-pure E0-semigroups of type II0.

Quantum Dynamical Semigroups and Decoherence

We prove a version of the Jacobs‐de Leeuw‐Glicksberg splitting theorem for weak* continuous one‐parameter semigroups on dual Banach spaces. This result is applied to give sufficient conditions for a quantum dynamical semigroup to display decoherence. The underlying notion of decoherence is that introduced by Blanchard and Olkiewicz (2003). We discuss this notion in some detail.

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.

Linearization models for parabolic dynamical systems via Abel's functional equation

We study linearization models for continuous one-parameter semigroups of para- bolic type. In particular, we introduce new limit schemes to obtain solutions of Abel's functional equation and to study asymptotic behavior of such semigroups. The crucial point is that these solutions are univalent functions convex in one direction. In a parallel direction, we find analytic conditions which determine certain geometric properties of those functions, such as the location of their images in either a half-plane or a strip, and their containing either a half-plane or a strip. In the context of semigroup theory these geometric questions may be interpreted as follows: is a given one-parameter continuous semigroup either an outer or an inner conjugate of a group of automor- phisms? In other words, the problem is finding a fractional linear model of the semigroup which is defined by a group of automorphisms of the open unit disk. Our results enable us to establish some new important analytic and geometric characteristics of the asymptotic behavior of one-parameter continuous semigroups of holomorphic mappings, as well as to study the problem of existence of a backward flow invariant domain and its geometry.

The group of isometries of a Banach space and duality

We construct an example of a real Banach space whose group of surjective isometries has no uniformly continuous one-parameter semigroups, but the group of surjective isometries of its dual contains infinitely many of them. Other examples concerning numerical index, hermitian operators and dissipative operators are also shown.

On [formula omitted]-algebra of a semigroup of partial isometries

A C ∗ -algebra associated to strongly continuous one-parameter semigroups of partial isometries is introduced as a groupoid C ∗ -algebra C ∗ ( G ) . A one-to-one correspondence between non-degenerate representations of C ∗ ( G ) and the semigroups is established. It is proved also that C ∗ ( G ) contains an ideal J isomorphic to C 0 ( ( 0 , + ∞ ] , K ) and C ∗ ( G ) / J ≃ C 0 ( R ) , where K is the algebra of compact operators. The C ∗ -algebra C ∗ ( G ) is shown to be generated by one unbounded affiliated element whose image under each non-degenerate representation is the infinitesimal generator of the corresponding semigroup.

Weyl Transforms, the Heat Kernel and Green Function of a Degenerate Elliptic Operator

We give a formula for the heat kernel of a degenerate elliptic partial differential operator L on ℝ2 related to the Heisenberg group. The formula is derived by means of pseudo-differential operators of the Weyl type, {i.e.}, Weyl transforms, and the Fourier–Wigner transforms of Hermite functions, which form an orthonormal basis for L2(ℝ2). Using the heat kernel, we give a formula for the Green function of L. Applications to the global hypoellipticity of L in the sense of tempered distributions, the ultracontractivity and hypercontractivity of the strongly continuous one-parameter semigroup e−tL, t > 0, are given.

Superstable semigroups of operators

In [NR] the authors introduced the notion of superstable operators on a Banach space E using ultrapowers E u of E. In [HR] this notion was extended to strongly continuous one-parameter semigroups again by means of ultrapowers. It is the aim of the present paper to give an equivalent intrinsic definition of superstability (without the reference to ultrapowers). This definition allows us to improve the results of [NR] as well as of [HR]. We apply our results to semigroups of positive linear operators on Banach lattices and C ∗- algebras , respectively.