2013/09/24 by Marat Pliev, Pliev, Marat, Б.Б. Тасоев +2 · 1 citation
Mathematics · #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #Holomorphic and Operator Theory #math.FA #msc:47H30 #msc:47H99
paper · pdf · doi:10.48550/arxiv.1309.6074
arxiv created 2013/09/24 · arxiv updated 2013/09/25
The aim of this note is to introduce the space DU(V,W) of the dominated Uryson operators on lattice-normed spaces. We prove an "Up-and-down" type theorem for a positive abstract Uryson operator defined on a vector lattice and taking values in a Dedekind complete vector lattice. This result we apply to prove the decomposability of the lattice valued norm of the space DU(V,W) of all dominated Uryson operators. We obtain that, for a lattice-normed space V and a Banach-Kantorovich space W the space DU(V,W) is also a Banach-Kantorovich space. We prove that under some mild conditions, a dominated Uryson operator has an exact dominant. We obtain formulas for calculating the exact dominant of a dominated Uryson operator. We also consider laterally continuous and completely additive dominated Uryson operators and prove that a dominated Uryson operator is laterally continuous (completely additive) if and only if so is its exact dominant.