Algebraic and analytic properties of semigroups related to fixed point properties of non-expansive mappings

Expansive operators which are power bounded or algebraic

Given Hilbert space operators $P,T\in B(\H), P\geq 0$ invertible, $T$ is $(m,P)-$ expansive (resp., $(m,P)-$ isometric) for some positive integer $m$ if $\triangle_{T^*,T}^m(P)=\sum_{j=0}^m(-1)^j\left(\begin{array}{clcr}m\\j\end{array}\right){T^*}^jPT^j\leq 0$ (resp., $\triangle_{T^*,T}^m(P)=0$). An $(m,P)-$ expansive operator $T$ is power bounded if and only if it is a $C_{1\cdot}-$ operator which is similar to an isometry and satisfies $\triangle_{T^*,T}^n(Q)=0$ for some positive invertible operator $Q\in B(\H)$ and all integers $n\geq 1$. If, instead, $T$ is an algebraic $(m,I)-$ expansive …

Semigroups associated with differential-algebraic equations

We consider differential-algebraic equations in infinite dimensional state spaces and study, under which conditions we can associate a $C_{0}$-semigroup with such equations. We determine the right space of initial values and characterise the existence of a $C_{0}$-semigroup in the case of operator pencils with polynomially bounded resolvents.

抽象微分代数方程的积分半群

Integrated Semigroups for abstract differential-algebraic equations

本研究将抽象柯西问题的积分半群概念拓展到具有预解指数的无穷维微分代数方程（DAEs）场景，构建的框架可用于分析非齐次DAEs并刻画其解。

Positive semigroups and algebraic Riccati equations in Banach spaces

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, …

Algebraic structure of semigroup compactifications: Pym's and Veech's Theorems and strongly prime points

The spectrum of an admissible subalgebra $\mathscr{A}(G)$ of $\mathscr{LUC}(G)$, the algebra of right uniformly continuous functions on a locally compact group $G$, constitutes a semigroup compactification $G^\mathscr{A}$ of $G$. In this paper we analyze the algebraic behaviour of those points of $G^\mathscr{A}$ that lie in the closure of $\mathscr{A}(G)$-sets, sets whose characteristic function can be approximated by functions in $\mathscr{A}(G)$. This analysis provides a common ground for far reaching generalizations of Veech's property (the action of $G$ on $G^\mathscr{LUC}$ is free) and Py…

Pontryagin Space Structure in Reproducing Kernel Hilbert Spaces over *-semigroups

The geometry of spaces with indefinite inner product, known also as Krein spaces, is a basic tool for developing Operator Theory therein. In the present paper we establish a link between this geometry and the algebraic theory of *-semigroups. It goes via the positive definite functions and related to them reproducing kernel Hilbert spaces. Our concern is in describing properties of elements of the semigroup which determine shift operators which serve as Pontryagin fundamental symmetries