Generator Of Semigroup Research Articles

The Induced Semigroup of Schwarz Maps to the Space of Hilbert-Schmidt Operators

We prove that for every semigroup of Schwarz maps on the von~Neumann algebra of all bounded linear operators on a Hilbert space which has a subinvariant faithful normal state there exists an associated semigroup of contractions on the space of Hilbert-Schmidt operators of the Hilbert space. Moreover, we show that if the original semigroup is weak$^*$ continuous then the associated semigroup is strongly continuous. We introduce the notion of the extended generator of a semigroup on the bounded operators of a Hilbert space with respect to an orthonormal basis of the Hilbert space. We describe the form of the generator of a quantum Markov semigroup on the von~Neumann algebra of all bounded linear operators on a Hilbert space which has an invariant faithful normal state under the assumption that the generator of the associated semigroup has compact resolvent.

Probability inequalities and tail estimates for metric semigroups

We study probability inequalities leading to tail estimates in a general semigroup $\mathscr{G}$ with a translation-invariant metric $d_{\mathscr{G}}$. (An important and central example of this in the functional analysis literature is that of $\mathscr{G}$ a Banach space.) Using our prior work [Ann. Prob. 2017] that extends the Hoffmann-Jorgensen inequality to all metric semigroups, we obtain tail estimates and approximate bounds for sums of independent semigroup-valued random variables, their moments, and decreasing rearrangements. In particular, we obtain the "correct" universal constants in several cases, extending results in the Banach space literature by Johnson-Schechtman-Zinn [Ann. Prob. 1985], Hitczenko [Ann. Prob. 1994], and Hitczenko and Montgomery-Smith [Ann. Prob. 2001]. Our results also hold more generally, in a very primitive mathematical framework required to state them: metric semigroups $\mathscr{G}$. This includes all compact, discrete, or (connected) abelian Lie groups.

Almost automorphy and Veech's relations of dynamics on compact T2-spaces

Let (T,X) be a dynamics on a compact T2-space X with phase semigroup T and with phase transformation T×X→(t,x)↦txX. Define two relations V and Q in X byV[x]={y|∃yn→y,tn∈T,z∈Xs.t.tnyn=z,tnx→z} andQ[x]={y|∃(xn,yn)→(x,y),tn∈Ts.t.lim⁡tnxn=lim⁡tnyn}. A point x of X is “almost automorphic” (a.a.) iff V[x]={x}. We study the a.a. dynamics and Veech's relation V of (T,X). We show if (T,X) is minimal with T a separable amenable group, then V[x]=Q[x] for x∈X; it is a partial solution to a question of Veech [35]. Moreover, using an example we show V[x]‾≠Q[x] for T in general semigroups; this solves the semigroup version of a question of Veech [36] and Auslander, Greschonig and Nagar [3]. In particular, we give a complete proof of Veech's Principal Structure Theorem for a.a. flows of any group.

Свободные прямоугольные n-кратные полугруппы

n-кратной полугруппой называется непустое множество G, снабженное n бинарными операциями $$\fbox{1}\,, \fbox{2}\,, ..., \fbox{n}\,,$$ удовлетворяющими аксиомам $$(x\fbox{r} \, y) \fbox{s}\, z=x\fbox{r}\,(y\fbox{s}\,z)$$ для всех $$x,y,z \in G$$ и $$r,s\in \{1,2,...,n\}.$$ Это понятие рассматривал Н.А.Корешков в контексте теории n-кратных алгебр ассоциативного типа. Доппельполугруппы являются 2-кратными полугруппами. n-кратные полугруппы имеют связи с интерассоциативными полугруппами, димоноидами, триоидами, доппельалгебрами, дуплексами, G-димоноидами и рестриктивными биполугруппами. Если операции n-кратной полугруппы совпадают, то она превращается в полугруппу. Таким образом, n-кратные полугруппы являются обобщением полугрупп. Класс всех n-кратных полугрупп образует многообразие. Недавно были построены свободная n-кратная полугруппа, свободная коммутативная n-кратная полугруппа, свободная k-нильпотентная n-кратная полугруппа и свободное произведение произвольных n-кратных полугрупп. Класс всех прямоугольных n-кратных полугрупп, то есть n-кратных полугрупп с n прямоугольными полугруппами, образует подмногообразие многообразия n-кратных полугрупп. В этой статье мы строим свободную прямоугольную n-кратную полугруппу и характеризуем наименьшую прямоугольную конгруэнцию на свободной n-кратной полугруппе.

Functional Calculus for the Series of Semigroup Generators Via Transference

In this paper, apply an established transference principle to obtain the boundedness of certain functional calculi for the sequence of semigroup generators. It is proved that if be the sequence generates 0- semigroups on a Hilbert space, then for each the sequence of operators has bounded calculus for the closed ideal of bounded holomorphic functions on right half–plane. The bounded of this calculus grows at most logarithmically as. As a consequence decay at ∞. Then showed that each sequence of semigroup generator has a so-called (strong) m-bounded calculus for all m∈ℕ, and that this property characterizes the sequence of semigroup generators. Similar results are obtained if the underlying Banach space is a UMD space. Upon restriction to so-called semigroups, the Hilbert space results actually hold in general Banach spaces.

Spectral inequality and resolvent estimate for the bi-Laplace operator

On a compact Riemannian manifold with boundary, we prove a spectral inequality for the bi-Laplace operator in the case of so-called “clamped” boundary conditions, that is, homogeneous Dirichlet and Neumann conditions simultaneously.We also prove a resolvent estimate for the generator of the damped plate semigroup associated with these boundary conditions. The spectral inequality allows one to observe finite sums of eigenfunctions for this fourth-order elliptic operator, from an arbitrary open subset of the manifold. Moreover, the constant that appears in the inequality grows as exp (C \mu^{1/4}) where \mu is the largest eigenvalue associated with the eigenfunctions appearing in the sum. This type of inequality is known for the Laplace operator. As an application, we obtain a null-controllability result for a higher-order parabolic equation. The resolvent estimate provides the spectral behavior of the plate semigroup generator on the imaginary axis. This type of estimate is known in the case of the damped wave semigroup. As an application, we deduce a stabilization result for the damped plate equation, with a log-type decay. The proofs of both the spectral inequality and the resolvent estimate are based on the derivation of different types of Carleman estimates for an elliptic operator related to the bi-Laplace operator, in the interior and at some boundaries. One of these estimates exhibits a loss of one full derivative. Its proof requires the introduction of an appropriate semi-classical calculus and a delicate microlocal argument.

Norm Estimates for a Semigroup Generated by the Sum of Two Operators with an Unbounded Commutator

Let A be the generator of an analytic semigroup $$(e^{At})_{t\ge 0}$$ on a Banach space $${\mathcal {X}}$$, B be a bounded operator in $${\mathcal {X}}$$ and $$K=AB-BA$$ be the commutator. Assuming that there is a linear operator S having a bounded left-inverse operator $$S_l^{-1}$$, such that $$\int _{0}^{\infty } \Vert Se^{At}\Vert \Vert e^{Bt}\Vert dt<\infty $$, and the operator $$KS_l^{-1}$$ is bounded and has a sufficiently small norm, we show that $$\int _{0}^{\infty } \Vert e^{(A+B)t}\Vert dt<\infty $$, where $$(e^{(A+B)t})_{t\ge 0}$$ is the semigroup generated by $$A+B$$. In addition, estimates for the supremum- and $$L^1$$-norms of the difference $$e^{(A+B)t}-e^{At}e^{Bt}$$ are derived.

Exponential sensitivity and turnpike analysis for linear quadratic optimal control of general evolution equations

We analyze the sensitivity of linear quadratic optimal control problems governed by general evolution equations with bounded or admissible control operator. We show, that if the problem is stabilizable and detectable, the solution of the extremal equation can be bounded by the right-hand side including initial data with the bound being independent of the time horizon. Consequently, the influence of perturbations of the extremal equations decays exponentially in time. This property can for example be used to construct efficient space and time discretizations for a Model Predictive Control scheme. Furthermore, a turnpike property for unbounded but admissible control of general semigroups can be deduced.

On the energy decay rates for the 1D damped fractional Klein–Gordon equation

AbstractWe consider the fractional Klein–Gordon equation in one spatial dimension, subjected to a damping coefficient, which is non‐trivial and periodic, or more generally strictly positive on a periodic set. We show that the energy of the solution decays at the polynomial rate for and at some exponential rate when . Our approach is based on the asymptotic theory of C0 semigroups in which one can relate the decay rate of the energy in terms of the resolvent growth of the semigroup generator. The main technical result is a new observability estimate for the fractional Laplacian, which may be of independent interest.

An analytic semigroup generated by a fractional differential operator

We study a space-fractional diffusion problem, where the non-local diffusion flux involves the Caputo derivative of the diffusing quantity. We prove the unique existence of regular solutions to this problem by means of the semigroup theory. We show that the operator defined as divergence of product of a positive function with the Caputo derivative is a generator of analytic semigroup.

A Besov algebra calculus for generators of operator semigroups and related norm-estimates

We construct a new bounded functional calculus for the generators of bounded semigroups on Hilbert spaces and generators of bounded holomorphic semigroups on Banach spaces. The calculus is a natural (and strict) extension of the classical Hille–Phillips functional calculus, and it is compatible with the other well-known functional calculi. It satisfies the standard properties of functional calculi, provides a unified and direct approach to a number of norm-estimates in the literature, and allows improvements of some of them.

Geometric properties of the nonlinear resolvent of holomorphic generators

Let f be the infinitesimal generator of a one-parameter semigroup of holomorphic self-mappings of the open unit disk Δ. Our main purpose is to study properties of the family R of non-linear resolvents (I+rf)−1:Δ→Δ,r≥0, in the spirit of classical geometric function theory. To make a connection with this theory, we mostly consider the case where f(0)=0 and f′(0) is a positive real number. We found, in particular, that R forms an inverse Löwner chain of hyperbolically convex functions. Moreover, each element of R satisfies the Noshiro-Warschawski condition. This, in turn, implies that each element of R is also the infinitesimal generator of a one-parameter semigroup on Δ. We consider also quasiconformal extensions of elements of R. Finally we study the existence of repelling fixed points of this family.

Weak Poincar\xe9 Inequalities in the Absence of Spectral Gaps

For generators of Markov semigroups which lack a spectral gap, it is shown how bounds on the density of states near zero lead to a so-called weak Poincaré inequality (WPI), originally introduced by Liggett (Ann Probab 19(3):935–959, 1991). Applications to general classes of constant coefficient pseudodifferential operators are studied. Particular examples are the heat semigroup and the semigroup generated by the fractional Laplacian in the whole space, where the optimal decay rates are recovered. Moreover, the classical Nash inequality appears as a special case of the WPI for the heat semigroup.

An $$L^{p}$$-Approach to the Well-Posedness of Transport Equations Associated with a Regular Field: Part I

Transport equations associated with a Lipschitz field $$\mathscr {F}$$ on some subspace of $${\mathbb {R}}^N$$ endowed with some general measure $$\mu $$ are considered. Our aim is to extend the results obtained in two previous contributions (Arlotti et al. in Mediterr J Math 6:367–402, 2009, Mediterr J Math 8:1–35, 2011) in the $$L^{1}$$-context to $$L^{p}$$-spaces $$1< p <\infty $$. This is the first part of a two-part contribution (see in Arlotti and Lods An $$L^{p}$$-approach to the well-posedness of transport equations associated with a regular field—part II, Mediterr. J. Math. 16:145, 2019, for the second part) and we here establish the general mathematical framework we are dealing with and notably prove trace formula and uniqueness of boundary value transport problems with abstract boundary conditions. The abstract results of this first part will be used in the Part II of this work (Arlotti and Lods in Meditter J Math 16:145, 2019) to deal with general initial and boundary value problems and semigroup generation properties.

An $$L^{p}$$-Approach to the Well-Posedness of Transport Equations Associated to a Regular Field: Part II

Transport equations associated with a Lipschitz field $$\mathscr {F}$$ on some subspace of $${\mathbb {R}}^N$$ endowed with some general measure $$\mu $$ are considered. Our aim is to extend the results obtained in two previous contributions (Arlotti et al. in Mediterr J Math 6:367–402, 2009, Mediterr J Math 8:1–35, 2011) in the $$L^{1}$$-context to $$L^{p}$$-spaces $$1< p <\infty $$. This is the first part of a two-part contribution (see in Arlotti and Lods An $$L^{p}$$-approach to the well-posedness of transport equations associated with a regular field—part II, Mediterr. J. Math. 16:145, 2019, for the second part) and we here establish the general mathematical framework we are dealing with and notably prove trace formula and uniqueness of boundary value transport problems with abstract boundary conditions. The abstract results of this first part will be used in the Part II of this work (Arlotti and Lods in Meditter J Math 16:145, 2019) to deal with general initial and boundary value problems and semigroup generation properties.

Abundance of progressions in a commutative semigroup by elementary means

Furstenberg and Glasner proved that for an arbitrary k in N, any piecewise syndetic set contains k term arithmetic progression and such collection is also piecewise syndetic in Z. They used algebraic structure of beta N. The above result was extended for arbitrary semigroups by Bergelson and Hindman, again using the structure of Stone-Cech compactification of general semigroup. Beiglbock provided an elementary proof of the above result and asked whether the combinatorial argument in his proof can be enhanced in a way which makes it applicable to a more abstract setting. In this work we provide a positive answer to Beiglbock question for commutative semigroup.

The positive maximum principle on Lie groups

We extend a classical theorem of Courr\`{e}ge to Lie groups in a global setting, thus characterising all linear operators on the space of smooth functions of compact support that satisfy the positive maximum principle. We show that these are L\'{e}vy type operators (with variable characteristics), and pseudo--differential operators when the group is compact. If the characteristics are constant, then the operator is the generator of the contraction semigroup associated to a convolution semigroup of sub--probability measures.

On the theory of the Kolmogorov operator in the spaces $L^p$ and $C_\infty$

We obtain the basic results concerning the problem of constructing operator realizations of the formal differential expression $\nabla \cdot a \cdot \nabla - b \cdot \nabla$ with measurable matrix $a$ and vector field $b$ having critical-order singularities as the generators of Markov semigroups in $L^p$ and $C_\infty$.

Open-quantum-system dynamics: Recovering positivity of the Redfield equation via the partial secular approximation

We show how to recover complete positivity (and hence positivity) of the Redfield equation via a coarse grain average technique. We derive general bounds for the coarse graining time scale above which the positivity of the Redfield equation is guaranteed. It turns out that a coarse grain time scale has strong impact on the characteristics of the Lamb shift term and implies in general non-commutation between the dissipating and the Hamiltonian components of the generator of the dynamical semi-group. Finally we specify the analysis to a two-level system or a quantum harmonic oscillator coupled to a fermionic or bosonic thermal environment via dipole-like interaction.

Supercritical superprocesses: Proper normalization and non-degenerate strong limit

Suppose that $X=\{X_t, t\ge 0; \mathbb{P}_{\mu}\}$ is a supercritical superprocess in a locally compact separable metric space $E$. Let $\phi_0$ be a positive eigenfunction corresponding to the first eigenvalue $\lambda_0$ of the generator of the mean semigroup of $X$. Then $M_t:=e^{-\lambda_0t}\langle\phi_0, X_t\rangle$ is a positive martingale. Let $M_\infty$ be the limit of $M_t$. It is known (see, J. Appl. Probab. 46 (2009), 479--496) that $M_\infty$ is non-degenerate iff the $L\log L$ condition is satisfied. In this paper we are mainly interested in the case when the $L\log L$ condition is not satisfied. We prove that, under some conditions, there exist function $\gamma_t$ on $[0, \infty)$ and a non-degenerate random variable $W$ such that for any finite nonzero Borel measure $\mu$ on $E$, $$ \lim_{t\to\infty}\gamma_t\langle \phi_0,X_t\rangle =W,\qquad\mbox{a.s.-}\mathbb{P}_{\mu}. $$ We also give the almost sure limit of $\gamma_t\langle f,X_t\rangle$ for a class of general test functions $f$.