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Martin R. Weber

Publications and source records attributed to Martin R. Weber.

5 recordsLinked to original sources
arXiv2020

Almost Interior Points in Ordered Banach Spaces and the Long--Term Behaviour of Strongly Positive Operator Semigroups

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 the theory presented in the first part of the paper we deduce sufficiency criteria for such semigroups to converge (strongly or in operator norm) as time tends to infinity. This generalises known results for semigroups on Banach lattices as well as on normally ordered Banach spaces with unit.

math.FA↗
arXiv2019

On $C$-compact orthogonally additive operators

We consider $C$-compact orthogonally additive operators in vector lattices. After providing some examples of $C$-compact orthogonally additive operators on a vector lattice with values in a Banach space we show that the set of those operators is a projection band in the Dedekind complete vector lattice of all regular orthogonally additive operators. In second part of the article we introduce a new class of vector lattices, called $C$-complete, and show that any laterally-to-norm continuous $C$-compact orthogonally additive operator from a $C$-complete vector lattice to a Banach space is narrow, which generalizes a result of Pliev and Popov.

math.FA↗
arXiv2019

On the Asymptotic Behaviour of some Positive Semigroups

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 bounded $C_0$-semigroup of linear continuous operators $(S_t)_{t\geq 0}$ on a Banach space contains a compact operator and the trajectories of the non-zero vectors $x\in K$ have the property from above then, in particular, $(S_t)$ and $(S^*_t)$ converge to their limit operator with repsect to the operator norm, respectively (Theorem 4). For weakly compact Markov operators in the space of real continuous functions on a compact topological space a corresponding result can be derived, that characterizes the long-term behaviour of regular Markov chains.

math.FA↗
arXiv2018

On finite elements in $f$-algebras and in product algebras

Finite elements, which are well-known and studied in the framework of vector lattices, are investigated in $\ell$-algebras, preferably in $f$-algebras, and in product algebras. The additional structure of an associative multiplication leads to new questions and some new properties concerning the collections of finite, totally finite and self-majorizing elements. In some cases the order ideal of finite elements is a ring ideal as well. It turns out that a product of elements in an $f$-algebra is a finite element if at least one factor is finite. If the multiplicative unit exists, the latter plays an important role in the investigation of finite elements. For the product of certain $f$-algebras an element is finite in the algebra if and only if its power is finite in the product algebra.

math.FA↗