Martin R. Weber

Disjointness and order projections in the vector lattices of abstract Uryson operators

Projections onto several special subsets in the Dedekind complete vector lattice of orthogonally additive, order bounded (called abstract Uryson) operators between two vector lattices E and F are considered and some new formulas are provided.

On finite and totally finite elements in vector lattices

AbstractДоказывается, что в ар химедовой векторной структуреX для того, чтобы каждый финитный элем ент являлся также тот ально финитным, достаточно, чтобы топология в подпрост ранствеMΦ пространст ва всех максимальныхl-идеало вM векторной структурыX обладала тем свойст вом, что замыкание (вM) счетно го объединения базисны х множеств лежитвMΦ.

Finite elements in vector lattices

This book is the first systematical treatment of the theory of finite elements in Archimedean vector lattices and contains the results known on this topic up to the year 2013. It joins all important contributions achieved by a series of mathematicians that can only be found in scattered in literature.

On a maximum principle for inverse monotone matrices

Abstract Let γ = (γ1, …, γn) be a given vector with positive coordinates. A matrix A is said to satisfy the γ-maximum principle (γMP) if Ax = y, y ≥ 0 imply x ≥ 0 and max 1≤i≤n γ i χ i = max i ∈ N = (y) γ i χ i where N+) (y) is the set of indices such that y is positive. For an invertible matrix A with positive inverse the γMP is characterized geometrically by means of the behavior under A−1 of convex boundary parts of the simplex generated in ( R n+ by permissible multiples of the unit coordinate vectors. Some sufficient conditions and applications to M-matrices are given.

On matrices satisfying a maximum principle with respect to a cone

Abstract A matrix equation A x = y is considered in the space R n that is ordered by a cone K. In case K = R n + it is known that a matrix A is said to satisfy the maximum principle if A is invertible and for each y ∈ R n + ⧹ {0} the solution x belongs to R n+ and is such that xi = max1 ⩽ k ⩽ nxk and yi > 0 for some i. This concept is generalized to finitely generated and to circular cones K ⊂ R n . This is achieved by “evaluating” x and y with the aid of elements of a given base for the polar cone K∘ of K. The maximum principle is characterized geometrically by means of the behavior under A−1 of convex boundary parts of a base for K. A weighted maximum principle is investigated and an infinite dimensional example is indicated.

Finite elements in some vector lattices of nonlinear operators

We study the collection of finite elements

On finite elements in vector lattices and Banach lattices

\Phi _{1}\big ({\mathcal {U}}(E,F)\big )

ON POSITIVE INVERTIBILITY AND SPLITTINGS OF OPERATORS IN ORDERED BANACH SPACES

Φ1(U(E,F)) in the vector lattice