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 $σ(A)$.
The purpose of this paper is to revisit previous works of the author with J. Sjöstrand (2010--2021) proved in the Hilbert case by considering the Banach case at the light of a paper by Y.~Latushkin and V.~Yurov (2013).
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.
We give a relation between the exponential stability of $ C_{0}- $semigroup $ \textbf{T}=\left\lbrace T(t) \right\rbrace_{t\geq 0} $ and the solutions of Lyapunov inequality \( \left\langle QAx,x\right\langle +\left\langle Qx,Ax\right\langle \leq -||x||^{2}, \) in $ B^{+}(X,X^{*}) $, with $ X $ is a Banach space. The solutions of this inequality characterizes, the boundedness of the resolvent $ R(λ,A) $ inside and outside of the left half-plane $ \Reλ\geq 0 $, and also the left invertibility of the $ C_{0}- $semigroup T.
In this paper, we study the decay rate of the Cayley transform of the generator of a polynomially stable $C_0$-semigroup. To estimate the decay rate of the Cayley transform, for polynomial stability. Using this integral condition, we relate polynomial stability to Lyapunov equations. We also study robustness of polynomial stability for a certain class of structured perturbations.
Feynman-Kac semigroups appear in various areas of mathematics: non-linear filtering, large deviations theory, spectral analysis of Schrodinger operators among others. Their long time behavior provides important information, for example in terms of ground state energy of Schrodinger operators, or scaled cumulant generating function in large deviations theory. In this paper, we propose a simple and natural extension of the stability of Markov chains for these non-linear evolutions. As other classical ergodicity results, it relies on two assumptions: a Lyapunov condition that induces some compact…
We stretch the spectral bound equal growth bound condition along with a generalized Lyapunov stability theorem, known to hold for $C_0$-semigroups of normal operators on complex Hilbert spaces, to $C_0$-semigroups of scalar type spectral operators on complex Banach spaces. For such semigroups, we obtain exponential estimates with the best stability constants. We also extend to a Banach space setting a celebrated characterization of uniform exponential stability for $C_0$-semigroups on complex Hilbert spaces and thereby acquire a characterization of uniform exponential stability for scalar type…
The paper deals with the long-term behavior of positive operator semigroups on spaces of bounded functions and of signed measures, which have applications to parabolic equations with unbounded coefficients and to stochastic analysis. The main results are a Tauberian type theorem characterizing the convergence to equilibrium of strongly Feller semigroups and a generalization of a classical convergence theorem of Doob. None of these results requires any kind of time regularity of the semigroup.
Use of bundles of locally convex spaces in problems of convergence of semigroups of operators defined on different Banach spaces. Applications to spectral stability problems
In this paper, we study the positivity and (uniform) exponential stability of a large class of perturbed semigroups. Our approach is essentially based on the feedback theory of infinite-dimensional linear systems. The obtained results are applied to the stability of hyperbolic systems including those with a delay at the boundary conditions.
The well-known Batty's theorem states that if a $C_0$-semigroup $T(t)$ is bounded and the spectrum of the generator $A$ is contained in the open left-half plane of $\mathbb{C}$, then $\|T(t)A^{-1}\|$ tends to $0$. This can be thought of as a particular case of a more general property that, for $ω_0>-\infty$ and $(ω_0+i\mathbb{R})\cap σ(A)=\emptyset$ it holds $\|T(t)(A-ω_0 I)^{-1}\|/\|T(t)\|$ tends to 0. We show that it is true for $\|T(t)\|$ regular enough, however we give examples of unbounded semigroups, with the spectrum of the generator not contained in the open left-half plane of $\mathbb…
The role of parameter Jacobians in the stability of network outputs
University at Albany, State University of New York（纽约州立大学奥尔巴尼分校）；The University of Iowa（爱荷华大学）；Southern Illinois University Edwardsville（南伊利诺伊大学爱德华兹维尔分校）；Mathematical Reviews（数学评论）
A universal example for quantitative semi-uniform stability
University of Twente（特温特大学）；Leibniz Universität Hannover（汉诺威莱布尼茨大学）；University of Salzburg（萨尔茨堡大学）；University of York（约克大学）；Technische Universität Bergakademie Freiberg（弗莱贝格矿业科技大学）
