Related papers: Stability criteria for positive semigroups on orde…

Assume that a block operator of the form $\left(\begin{smallmatrix}A_{1}\\A_{2}\quad 0\end{smallmatrix}\right)$, acting on the Banach space $X_{1}\times X_{2}$, generates a contraction $C_{0}$-semigroup. We show that the operator $A_{S}$…

We give a short and elementary proof of the theorem of Lutz Weis that the growth bound of a positive $C_0$-semigroup on $L_p(\mu)$ equals the spectral bound of its generator. In addition, we generalise the result to the case of uniformly…

In this paper, we investigate well-posedness and stability properties of distributed parameter systems, with particular emphasis on linear positive control systems. We establish a characterization of the well-posedness in the Banach lattice…

In this paper we study the well-posedness of the evolution equation of the form $u'(t)=Au(t)+Cu(t)$, $t\ge 0$, where $A$ is the generator of a $C_0$- semigroup and $C$ is a (possibly unbounded) linear operator in a Banach space…

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…

We study rates of decay for $C_0$-semigroups on Banach spaces under the assumption that the norm of the resolvent of the semigroup generator grows with $\vert s\vert^{\beta}\log(\vert s\vert)^b$, $\beta, b \geq 0$, as $\vert…

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…

We study abstract sufficient criteria for open-loop stabilizability of linear control systems in a Banach space with a bounded control operator, which build up and generalize a sufficient condition for null-controllability in Banach spaces…

A basic result in semigroup theory states that every $C_0$-semigroup is quasi-contractive with respect to some appropriately chosen equivalent norm. This paper contains a counterpart of this well-known fact. Namely, by examining the…

In this paper the theory of evolution semigroups is developed and used to provide a framework to study the stability of general linear control systems. These include time-varying systems modeled with unbounded state-space operators acting…

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…

Let $A$ be an unbounded operator on a Banach space $X$. It is sometimes useful to improve the operator $A$ by extending it to an operator $B$ on a larger Banach space $Y$ with smaller spectrum. It would be preferable to do this with some…

We investigate for a bounded semigroup of linear operators $S$ on a Banach space $E$ and a vector $x \in E$, when relative compactness of $S(I-T)x$ for every $T \in S$ implies relative compactness of the orbit $Sx$. In particular, we derive…

We systematize and generalize recent results of Gerlach and Gl\"uck 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…

We characterise contractivity, boundedness and polynomial boundedness for a C_0-semigroup on a Banach space in terms of its cogenerator V (or the Cayley transform of the generator) or its resolvent. In particular, we extend results of…

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…

Let $T:X\to X$ be a linear power bounded operator on Banach space. Let $X_0$ is a subspace of vectors tending to zero under iterating of $T$. We prove that if $X_0$ is not equal to $X$ then there exists $\lambda$ in Sp(T) such that, for…

In this work, we revisit the Krein-Rutman theory for semigroups of positive operators in a Banach lattice framework and we provide some very general, efficient and handy results with constructive estimates about: the existence of a solution…

We present a new method for constructing $C_0$-semigroups for which properties of the resolvent of the generator and continuity properties of the semigroup in the operator-norm topology are controlled simultaneously. It allows us to show…

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…