A. Bátkai, B. Jacob, J. Voigt, J. Wintermayr
We consider positive perturbations of positive semigroups on AM-spaces and prove a result which is the dual counterpart of a famous perturbation result of Desch in AL-spaces. As an application we consider unbounded perturbations of the shift semigroup.
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We prove a perturbation result for positive semigroups, thereby extending a heat kernel estimate by Barlow, Grigor'yan and Kumagai for Dirichlet forms (\cite{bgk2009}) to positive semigroups. This also leads to a generalization of domination for semigroups on $L_p$-spaces.
For a given $C_0$-semigroup $(T(t))_{t\geq 0}$ we consider Staffans-Weiss perturbations $(B,C)$ of its generator as studied in [1] and investigate the robustness of asymptotic properties of the perturbed $C_0$-semigroup $(T_{BC}(t))_{t\geq 0}$. As a concrete application we study the asymptotic behavior of a neutral semigroup.
We discuss positive Miyadera-Voigt type perturbations for bi-continuous semigroups on AL-spaces with an additional locally convex topology generated by additive seminorms. Our main example is the space of bounded Borel measures.
In this paper, we consider positive Desch-Schappacher perturbations of bi-continuous semigroups on $\mathrm{AM}$-spaces with an additional property concerning the additional locally convex topology. As an examples, we discuss perturbations of the left-translation semigroup on the space of bounded continuous function on the real line and on the space of bounded linear operators.
We present a perturbation result for generators of $C_0$-semigroups which can be considered as an operator theoretic version of the Weiss-Staffans perturbation theorem for abstract linear systems. The result are illustrated by applications to the Desch-Schappacher, the Miyadera Voigt perturbation theorems, and to unbounded perturbations of the boundary conditions of a generator.
We prove a Desch-Schappacher type perturbation theorem for one-parameter semigroups on Banach spaces which are not strongly continuous for the norm, but possess a weaker continuity property. In this paper we chose to work in the framework of bi-continuous semigroups. This choice has the advantage that we can treat in a unified manner two important classes of semigroups: implemented semigroups on the Banach algebra $\mathscr{L}(E)$ of bounded, linear operators on a Banach space $E$, and semigroups on the space of bounded and continuous functions over a Polish space induced by jointly continuous…
We prove a Desch-Schappacher type perturbation theorem for strongly continuous and locally equicontinuous one-parameter semigroups which are defined on a sequentially complete locally convex space.
We present a generation theorem for positive semigroups on an $L^1$ space. It provides sufficient conditions for the existence of positive and integrable solutions of initial-boundary value problems. An application to a two-phase cell cycle model is given.
We consider eventually positive operator semigroups and study the question whether their eventual positivity is preserved by bounded perturbations of the generator or not. We demonstrate that eventual positivity is not stable with respect to large positive perturbation and that certain versions of eventual positivity react quite sensitively to small positive perturbations. In particular we show that if eventual positivity is preserved under arbitrary positive perturbations of the generator, then the semigroup is positive. We then provide sufficient conditions for a positive perturbation to pre…
Quasi-compactness and uniform stabilization on general Banach spaces under $θ$-subordinate perturbations of semigroup generators
本文证明在一般Banach空间上，解析半群在沿轨道紧的$\ heta$-从属扰动下保持拟紧性和一致指数稳定性，并推广至Crandall-Pazy半群及立即可微半群，最后通过反例说明$A$-有界扰动下稳定性可能缺失。
We describe a construction of cocyclic perturbations of the semigroup of shifts on the semiaxis by means of the theory of model spaces. It is shown that one can choose an inner function that determines the model space so that the elements of the perturbed semigroup have a prescribed spectral type and differ from the elements of the initial semigroup by operators from the Schatten-von Neumann class $\mathfrak{S}_p$, $p>1$. The case of the trace class $\mathfrak{S}_1$ perturbations is considered separately.
Let $Ω$ be an operator semigroup with generator $A$ in a sequentially complete locally convex topological vector space $E$. For a semigroup with generator $A+D$, where $D$ is a bounded linear operator on $E$, two integral equations are derived. A theorem on continuous dependence of a semigroup on its generator is proved. An application to random walk on $\mathbb{Z}$ is given.
We give a characterization of a variation of constants type estimate relating two positive semigroups on (possibly different) $L_p$-spaces to one another in terms of corresponding estimates for the respective generators and of estimates for the respective resolvents. The results have applications to kernel estimates for semigroups induced by accretive and non-local forms on $σ$-finite measure spaces.
In a recent paper we presented a general perturbation result for generators of $C_0$-semigroups. The aim of the present paper is to replace, in case the unperturbed semigroup is analytic, the various conditions appearing in this result by simpler assumptions on the domain and range of the operators involved. The power of our result to treat classes of PDE's systematically is illustrated by considering a generic example, a degenerate differential operator with generalized Wentzell boundary conditions and a reaction diffusion equation subject to Neumann boundary conditions with distributed unbou…
In this paper the Jessen's type inequality for normalized positive $C_0$-semigroups is obtained. An adjoint of Jessen's type inequality has also been derived for the corresponding adjoint-semigroup, which does not give the analogous results but the behavior is still interesting. Moreover, it is followed by some results regarding positive definiteness and exponential convexity of complex structures involving operators from a semigroup.
We consider positive operator semigroups on ordered Banach spac\-es and study the relation of their long time behaviour to two different domination properties. First, we analyse under which conditions almost periodicity and mean ergodicity of a semigroup $\mathcal{T}$ are inherited by other semigroups which are asymptotically dominated by $\mathcal{T}$. Then, we consider semigroups whose orbits asymptotically dominate a positive vector and show that this assumption is often sufficient to conclude strong convergence of the semigroup as time tends to infinity. Our theorems are applicable to time…
The paper is devoted to a linear dynamics for non-autonomous perturbation of the Gibbs semigroup on a separable Hilbert space. It is shown that evolution family {U(t, s)} 0$\le$s$\le$t solving the non-autonomous Cauchy problem can be approximated in the trace-norm topology by product formulae. The rate of convergence of product formulae approximants {U n (t, s)} {0$\le$s$\le$t,n$\ge$1} to the solution operator {U(t, s)} {0$\le$s$\le$t} is also established.
We give extensions of results on nonnegative matrix semigroups which deduce finiteness or boundedness of such semigroups from the corresponding local properties, e.g., from finiteness or boundedness of values of certain linear functionals applied to them. We also consider more general semigroups of functions.
The paper deals with multidimensional Bochner-Phillips functional calculus. In the previous paper by the author bounded perturbations of Bernstein functions of several commuting semigroup generators on Banach spaces where considered, conditions for Lipschitzness and estimates for the norm of commutators of such functions where proved. Also in the one-dimensional case the Frechet differentiability of Bernstein functions of semigroup generators on Banach spaces where proved and a generalization of Livschits-Kreĭn trace formula derived. The aim of the present paper is to prove the Frechet differe…
Stability of uniformly eventually positive $C_0$-semigroups on $L_p$-spaces
本文简明证明了$L_p(μ)$上正$C_0$-半群的增长界等于生成子谱界的Weis定理，并将该结果推广至一致最终正半群情形。
A new proof of a result of Lutz Weis is given, that states that the stability of a positive strongly continuous semigroup $(e^{tA})_{t \ge 0}$ on $L_p$ may be determined by the quantity $s(A)$. We also give an example to show that the dichotomy of the semigroup may not always be determined by the spectrum $σ(A)$.
In this paper we study the robustness of strong stability of a discrete semigroup on a Hilbert space under bounded finite rank perturbations. As the main result we characterize classes of perturbations preserving the strong stability of the semigroup.
We give a new and very short proof of a theorem of Greiner asserting that a positive and contractive $C_0$-semigroup on an $L^p$-space is strongly convergent in case that it has a strictly positive fixed point and contains an integral operator. Our proof is a streamlined version of a much more general approach to the asymptotic theory of positive semigroups developed recently by the authors. Under the assumptions of Greiner's theorem, this approach becomes particularly elegant and simple. We also give an outlook on several generalisations of this result.
We consider abstract equations of the form Ax=-z on a locally convex space, where A generates a positive semigroup and z is a positive element. This is an abstract version of the operator Lyapunov equation A*P+PA=-Q from control theory. It is proved that under suitable assumptions existence of a positive solution implies that -A has a positive inverse, and the generated semigroup is asymptotically stable. We do not require that z is an order unit, or that the space contains any order units. As an application, we generalize Wonham's theorem on the operator Lyapunov equations with detectable rig…
Domination of nonlinear semigroups generated by regular, local Dirichlet forms
本文刻画了局部非线性狄利克雷型半群支配另一泛函生成半群的充要条件，即后者为前者的特定零阶扰动，从而对介于诺伊曼与狄利克雷边界条件之间的扰动进行分类。
These are the notes of a lecture given at the university of Wroclaw in 1996. We present results of semigroups of (sub)positive contractions on L^p-spaces. The dilation theorem of Akcoglue and sucheston is considered as a starting point. We construct a dilation of a strongly continuous semigroup of (sub)positive contractions on L^p-space, 1<p<infinity by a strongly continuous group of invertible isometries acting on some larger L^p-space. The transference results, for functions of the generators, are established as well as some square and maximal functions. This applies especially to sub-markov…
Let S be a dense sub-semigroup of the positive real numbers, and let X be a separable, reflexive Banach space. This note contains a proof that every weakly continuous contractive semigroup of operators on X over S can be extended to a weakly continuous semigroup over the positive real numbers. We obtain similar results for non-linear, non-expansive semigroups as well. As a corollary we characterize all densely parametrized semigroups which are extendable to semigroups over the positive real numbers.
We present new conditions for semigroups of positive operators to converge strongly as time tends to infinity. Our proofs are based on a novel approach combining the well-known splitting theorem by Jacobs, de Leeuw and Glicksberg with a purely algebraic result about positive group representations. Thus we obtain convergence theorems not only for one-parameter semigroups but for a much larger class of semigroup representations. Our results allow for a unified treatment of various theorems from the literature that, under technical assumptions, a bounded positive $C_0$-semigroup containing or dom…
We present a new and very short proof of the fact that, for positive $C_0$-semigroups on spaces of continuous functions, the spectral and the growth bound coincide. Our argument, inspired by an idea of Vogt, makes the role of the underlying space completely transparent and also works if the space does not contain the constant functions - a situation in which all earlier proofs become technically quite involved. We also show how the argument can be adapted to yield the same result for semigroups that are only eventually positive rather than positive.
The perturbation semigroup was first defined in the case of $*$-algebras by Chamseddine, Connes and van Suijlekom. In this paper, we take $\mathcal{E}$ as a concrete operator system with unit. We first give a definition of gauge group $\mathcal{G}(\mathcal{E})$ of $\mathcal{E}$, after that we give the definition of perturbation semigroup of $\mathcal{E}$, and the closed perturbation semigroup of $\mathcal{E}$ with respect to the Haagerup tensor norm. We also show that there is a continuous semigroup homomorphism from the closed perturbation semigroup to the collection of unital completely boun…
The main purpose of this paper is to treat semigroups properties, like norm continuity, compactness and differentiability for perturbed semigroups in Banach spaces. In particular, we investigate three large classes of perturbations, Miyadera-Voigt, Desch-Schappacher and Staffans-Weiss perturbations. Our approach is mainly based on feedback theory of Salamon-Weiss systems. Our results are applied to abstract boundary integro-differential equations in Banach spaces.
On Structured Perturbations of Positive Semigroups
本文推广了正半群的扰动理论，给出Weiss-Staffans型结果，并应用于定义域扰动、边界反馈热方程及一阶导数扰动。
The first part of this article is a brief survey of the properties of so-called almost interior points in ordered Banach spaces. Those vectors can be seen as a generalization of ``functions which are strictly positive almost everywhere'' on $L^p$-spaces and of ``quasi-interior points'' in Banach lattices. In the second part we study the long--term behaviour of strongly positive operator semigroups on ordered Banach spaces; these are semigroups which, in a sense, map every non-zero positive vector to an almost interior point. Using the Jacobs--de Leeuw--Glicksberg decomposition together with th…
Locally Eventually Positive Operator Semigroups
本文建立 Banach 格上局部最终正算子半群理论，弱化最终正 $C_0$-半群对强正特征向量的要求，给出充分判据并拓展至 Laplace 平方与双 Laplace 等例子。
On the extension of one-parameter operator semigroups to completions of Archimedean vector lattices
本文研究了阿基米德向量格上单参数算子半群到其完备化的扩张，证明了在正半群类中到ru-完备化扩张的存在性与唯一性，并通过证明性质(R)向量格上的扩张定理，去除了相关结论中对ru-完备性的要求。
Motivated by B. Tsirelson's construction of $E_0$-semigroups of type III, we investigate a $C_0$-semigroup acting on the space of square integrable functions of the half line whose difference from the shift semigroup is a Hilbert-Schmidt operator. We give a description of such semigroups in terms of analytic functions on the right-half plane and construct several examples.
Order boundedness and order continuity properties of positive operator semigroups
本文在巴拿赫格框架下对比相对一致连续（ruc）半群与$C_0$-半群，证明ruc半群即短时间轨道序有界的正$C_0$-半群，并关联谱界与增长界相等性等三个相关研究主题。
We consider perturbations of dynamical semigroups on the algebra of all bounded operators in a Hilbert space generated by covariant completely positive measures on the semi-axis. The construction is based upon unbounded linear perturbations of generators of the preadjoint semigroups on the space of nuclear operators. As an application we construct a perturbation of the semigroup of non-unital *-endomorphisms on the algebra of canonical anticommutation relations resulting in the flow of shifts.
We develop a theory of eventually positive $C_0$-semigroups on Banach lattices, that is, of semigroups for which, for every positive initial value, the solution of the corresponding Cauchy problem becomes positive for large times. We give characterisations of such semigroups by means of spectral and resolvent properties of the corresponding generators, complementing existing results on spaces of continuous functions. This enables us to treat a range of new examples including the square of the Laplacian with Dirichlet boundary conditions, the bi-Laplacian on $L^p$-spaces, the Dirichlet-to-Neuma…
We perform an in-depth study of some domination and smoothing properties of linear operators and of their role within the theory of eventually positive operator semigroups. On the one hand we prove that, on many important function spaces, they imply compactness properties. On the other hand, we show that these conditions can be omitted in a number of Perron--Frobenius type spectral theorems. We furthermore prove a Kre\uın--Rutman type theorem on the existence of positive eigenvectors and eigenfunctionals under certain eventual positivity conditions.
We consider dynamical semigroups with unbounded Kossakowski-Lindblad-Davies generators which are related to evolution of an open system with a tuned repeated harmonic perturbation. Our main result is the proof of existence of uniquely determined minimal trace-preserving strongly continuous dynamical semigroups on the space of density matrices. The corresponding dual W *-dynamical system is shown to be unital quasi-free and completely positive automorphisms of the CCR-algebra. We also comment on the action of dynamical semigroups on quasi-free states.
If $(T_t)$ is a semigroup of Markov operators on an $L^1$-space that admits a non-trivial lower bound, then a well-known theorem of Lasota and Yorke asserts that the semigroup is strongly convergent as $t \to \infty$. In this article we generalise and improve this result in several respects. First, we give a new and very simple proof for the fact that the same conclusion also holds if the semigroup is merely assumed to be bounded instead of Markov. As a main result we then prove a version of this theorem for semigroups which only admit certain individual lower bounds. Moreover, we generalise a…
Stability of Individually Eventually Positive Semigroups on $L^p$-Spaces
该研究证明了L^p空间上个体最终正C0半群的增长界与其生成元的谱界一致，通过适配算子范围均匀化原理完成证明，解答了Vogt提出的相关问题。
In this paper we prove a Hille-Yosida type theorem for relatively uniformly continuous positive semigroups on vector lattices. We introduce the notions of relatively uniformly continuous, differentiable, and integrable functions on $\mathbb{R}_+$. These notions allow us to study the generators of relatively uniformly continuous semigroups. Our main result provides sufficient and necessary conditions for an operator to be the generator of an exponentially order bounded, relatively uniformly continuous, positive semigroup.
This article deals with a variation of constants type inequality for semigroups acting consistently on a scale of Banach spaces. This inequality can be characterized by a corresponding (easy to verify) inequality for their generators. The results have applications to heat kernel estimates and provide a unified perspective to estimates of these type. Moreover, bi-continuous semigroups can be treated as well.
Analyticity of positive semigroups is inherited under domination
本文研究 Banach 格上正 C_0-半群间的支配关系，借助向量值函数空间中乘法算子的正算子谱理论，证明被支配半群继承支配半群的解析性，解决 Arendt 于 2004 年提出的公开问题。
Smoothing of operator semigroups under relatively bounded perturbations
研究强连续算子半群的平滑性质，探讨其在相对有界扰动下的稳定性，推导出谱扰动定理，对椭圆算子驱动的演化方程长期动力学有影响。
We generalize Wonham's theorem on solvability of algebraic operator Riccati equations to Banach spaces, namely there is a unique stabilizing solution to A*P+PA-PBB*P+C*C=0 when (A,B) is exponentially stabilizable and (C,A) is exponentially detectable. The proof is based on a new approach that treats the linear part of the equation as the generator of a positive semigroup on the space of symmetric operators from a Banach space to its dual, and the quadratic part as an order concave map. A direct analog of global Newton's iteration for concave functions is then used to approximate the solution, …
We develop a systematic theory of eventually positive semigroups of linear operators mainly on spaces of continuous functions. By eventually positive we mean that for every positive initial condition the solution to the corresponding Cauchy problem is positive for large enough time. Characterisations of such semigroups are given by means of resolvent properties of the generator and Perron--Frobenius type spectral conditions. We apply these characterisations to prove eventual positivity of several examples of semigroups including some generated by fourth order elliptic operators and a delay dif…
We study perturbations $(\tildeτ_t)_{t\ge 0}$ of the semigroup of shifts $(τ_t)_{t\ge 0}$ on $L^2(\R_+)$ with the property that $\tildeτ_t - τ_t$ belongs to a certain Schatten-von Neumann class $\gS_p$ with $p\ge 1$. We show that, for the unitary component in the Wold-Kolmogorov decomposition of the cogenerator of the semigroup $(\tildeτ_t)_{t\ge 0}$, {\it any singular} spectral type may be achieved by $\gS_1$ perturbations. We provide an explicit construction for a perturbation with a given spectral type based on the theory of model spaces of the Hardy space $H^2$. Also we show that we may ob…
This paper is to study some conditions on semigroups, generated by some class of non-densely defined operators in the closure of its domain, in order that certain bounded perturbations preserve some regularity properties of the semigroup such as norm continuity, compactness, differentiability and analyticity. Furthermore, we study the critical and essential growth bound of the semigroup under bounded perturbations. The main results generalize the corresponding results in the case of Hille-Yosida operators. As an illustration, we apply the main results to study the asymptotic behaviors of a cla…
The classical Perron-Frobenius theory asserts that an irreducible matrix $A$ has cyclic peripheral spectrum and its spectral radius $r(A)$ is an eigenvalue corresponding to a positive eigenvector. In Radjavi (1999) and Radjavi and Rosenthal (2000), this was extended to semigroups of matrices and of compact operators on $L_p$-spaces. We extend this approach to operators on an arbitrary Banach lattice $X$. We prove, in particular, that if $\iS$ is a commutative irreducible semigroup of positive operators on $X$ containing a compact operator $T$ then there exist positive disjoint vectors $x_1,...…
We systematize and generalize recent results of Gerlach and Glück on the strong convergence and spectral theory of bounded (positive) operator semigroups $(T_s)_{s\in S}$ on Banach spaces (lattices). (Here, $S$ can be an arbitrary commutative semigroup, and no topological assumptions neither on $S$ nor on its representation are required.) To this aim, we introduce the "semigroup at infinity" and give useful criteria ensuring that the well-known Jacobs--de Leeuw--Glicksberg splitting theory can be applied to it. Next, we confine these abstract results to positive semigroups on Banach lattices w…
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 po…
The Damek-Ricci spaces are solvable Lie groups and noncompact harmonic manifolds. The rank one Riemannian symmetric spaces of noncompact type sits inside it as a thin subclass. In this note we establish that for any Damek-Ricci space $S$, the heat semigroup generated by certain perturbation of the Laplace-Beltrami operator is {\em chaotic} on the Lorentz spaces $L^{p,q}(S)$, $2<p<\infty, 1\le q<\infty$ and subspace-chaotic on the weak $L^p$-spaces. We show that both the amount of perturbation and the range of $p$ are sharp. This generalizes a result in \cite{J-W} which proves that under identi…
The eigenvectors of an ergodic semigroup of linear normal positive unital maps on a von Neumann algebra are described. Moreover, it is shown by means of examples, that mere positivity of the maps in question is not sufficient for Frobenius theory as in S. Albeverio and R. Høegh-Krohn, \emph{Frobenius theory of positive maps of von Neumann algebras}, Comm. Math. Phys. \bf{64}(1978), 83--94, to hold.
We prove some new characterisations of honesty of the perturbed semigroup in Kato's Perturbation Theorem on abstract state spaces via three approaches, namely mean ergodicity of operators, adjoint operators and uniqueness of the perturbed semigroup. We then apply Kato's Theorem on abstract state spaces and the honesty theory linked to it to the study of quantum dynamical semigroups. We show that honesty is the natural generalisation of the notion of conservativity.
Similar to the theory of finite Markov chains it is shown that in a Banach space $X$ ordered by a closed cone $K$ with nonempty interior int($K$) a power bounded positive operator $A$ with compact power such that its trajectories for positive vectors eventually flow into int($K$), defines a "limit distribution", i.e. its adjoint operator has a unique fixed point in the dual cone. Moreover, the sequence (A^n) converges with respect to the strong operator topology and for each functional $f\in X'$ the sequence $((A^*)^n(f))$ converges with respect to the weak*-topology (Theorem 5). If a positive…
In order to successfully explore quantum systems which are perturbations of simple models, it is essential to understand the complexity of perturbation bounds. We must ask ourselves: How quantum many-body systems can be artificially engineered to produce the needed behavior. Therefore, it is convenient to make use of abstract framework to better understand classical and quantum systems. Thus, our investigation's purpose is to explore stability and perturbation bounds of positive $C_0$-semigroups on abstract state spaces using the Dobrushin's ergodicity coefficient. Consequently, we obtain a li…
A Generalization of Lévy's Theorem on Positive Matrix Semigroups
本文推广了正矩阵半群中分量恒正或恒为零的 Lévy 定理，去除了马尔可夫性和时间零处连续性条件，并给出序列空间上的算子半群表述。
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