Domination problem for AM-compact abstract Uryson operators

The Boolean algebra of fragments of a positive abstract Uryson operator recently was described in M. Pliev (Positivity, doi:10.1007/s11117-016-0401-9, 2016). Using this result, we prove a theorem of domination for AM-compact positive abstract Uryson operators from a Dedekind complete vector lattice E to a Banach lattice F with an order continuous norm.     

Spectral order of operators and range projections

We study the effect algebra (i.e. the positive part of the unit ball of an operator algebra) and its relation to the projection lattice from the perspective of the spectral order. A spectral orthomorphism is a map between effect algebras which preserves the spectral order and orthogonality of elements. We show that if the spectral orthomorphism preserves the multiples of the unit, then it is a restriction of a Jordan homomorphism between the corresponding algebras. This is an optimal extension of the Dye's theorem on orthomorphisms of the projection lattices to larger structures containing the projections. Moreover, results on automatic countable additivity of spectral homomorphisms are proved. Further, we study the order properties of the range projection map, assigning to each positive contraction in a JBW algebra its range projection. It is proved that this map preserves infima of positive contractions in the spectral (respectively standard order) if, and only if, the projection lattice of the algebra in question is a modular lattice.     

Finite elements in some vector lattices of nonlinear operators

We study the collection of finite elements \(\Phi _{1}\big ({\mathcal {U}}(E,F)\big )\) in the vector lattice \({\mathcal {U}}(E,F)\) of orthogonally additive, order bounded (called abstract Uryson) operators between two vector lattices E and F, where F is Dedekind complete. In particular, for an atomic vector lattice E it is proved that for a finite element in \(\varphi \in {\mathcal {U}}(E,{\mathbb {R}})\) there is only a finite set of mutually disjoint atoms, where \(\varphi \) does not vanish and, for an atomless vector lattice the zero-vector is the only finite element in the band of \(\sigma \)-laterally continuous abstract Uryson functionals. We also describe the ideal \(\Phi _{1}\big ({\mathcal {U}}({\mathbb {R}}^n,{\mathbb {R}}^m)\big )\) for \(n,m\in {\mathbb {N}}\) and consider rank one operators to be finite elements in \({\mathcal {U}}(E,F)\).     

Order-bounded operators in vector lattices and in spaces of measurable functions

The survey is devoted to the presentation of the state of the art of a series of directions of the theory of order-bounded operators in vector lattices and in spaces of measurable functions. The theory of disjoint operators, the generalized Hewitt-Yosida theorem, the connection with p-absolutely summing operators are considered in detail.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 26, pp. 3–63, 1988.     

On some vector lattices of operators and their finite elements

If the vector space of all regular operators between the vector lattices E and F is ordered by the collection of its positive operators, then the Dedekind completeness of F is a sufficient condition for to be a vector lattice. and some of its subspaces might be vector lattices also in a more general situation. In the paper we deal with ordered vector spaces of linear operators and ask under which conditions are they vector lattices, lattice-subspaces of the ordered vector space or, in the case that is a vector lattice, sublattices or even Banach lattices when equipped with the regular norm. The answer is affirmative for many classes of operators such as compact, weakly compact, regular AM-compact, regular Dunford-Pettis operators and others if acting between appropriate Banach lattices. Then it is possible to study the finite elements in such vector lattices , where F is not necessary Dedekind complete. In the last part of the paper there will be considered the question how the order structures of E, F and are mutually related. It is also shown that those rank one and finite rank operators, which are constructed by means of finite elements from E′ and F, are finite elements in . The paper contains also some generalization of results obtained for the case in [10].      

On the notion of stability of order convergence in vector lattices

In the theory of Banach lattices the following criterion for a norm to be order continuous is established: a norm is order continuous if and only if every order bounded sequence of positive pairwise disjoint elements in a lattice converges to zero in norm. In this paper we give a criterion for order convergence to be stable in a rather wide class of vector lattices which includes all Köthe spaces. The formulation of the criterion is analogous to that of the above-mentioned criterion for a norm to be order continuous. Namely, under certain conditions imposed on a vector lattice, stability of order convergence is equivalent to the condition that every order bounded sequence of positive pairwise disjoint elements converges relatively uniformly to zero. Furthermore, we study some types of order ideals in vector lattices. In terms of these ideals we give clarified versions of the above-stated criterions. As for notation and terminology, see for example [1,2].Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 35, No. 5, pp. 1026–1031, September–October, 1994.     

On the modulus of disjointness preserving operators on complex vector lattices

Let L and M be Archimedean vector lattices such that and are complex vector lattices. We constructively and intrinsically prove that if is an order bounded disjointness preserving operator from into then the modulus

Capacity in abstract hilbert spaces and applications to higher order. differential operators

This paper introduces the notion of capacity in abstract Hilhertl spaces. It is proved that the spectral shift of aa self-adjoi at operator which is subjected to a, domain perturbation can he estimated in forms of this capacity. The results -are finally applied to higher-order partial differential operators     

Interpolation of nonlinear positive or order preserving operators on Banach lattices

Positivity - We study the relationship between exact interpolation spaces for positive, linear operators, for order preserving, Lipschitz continuous operators, and for positive...     

Theorems on decomposition of operators in L 1 and their generalization to vector lattices

The Rosenthal theorem on the decomposition for operators in L ₁ is generalized to vector lattices and to regular operators on vector lattices. The most general version turns out to be relatively simple, but this approach sheds new light on some known facts that are not directly related to the Rosenthal theorem. For example, we establish that the set of narrow operators in L ₁ is a projective component, which yields the known fact that a sum of narrow operators in L ₁ is a narrow operator. In addition to the Rosenthal theorem, we obtain other decompositions of the space of operators in L ₁, in particular the Liu decomposition. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 1, pp. 26–35, January, 2006.     

Bands in lattices of operators

We consider the lattice of regular operators on a Dedekind complete Banach lattice. We show that in general the projection onto a band generated by a lattice homomorphism need not be continuous and that the principal bands need not be closed for the operator norm. In fact it is possible to find a convergent sequence of operators all the members of which are disjoint from the limit.

     

On projections of arbitrary lattices

In this paper we prove that given any two point lattices _Λ1⊂Rⁿ${\Lambda }_1\subset {\mathbb{R}}^n$ and _Λ2⊂R^n−k${\Lambda }_2\subset {\mathbb{R}}^{n-k}$, there is a set of k   vectors {_v1,…,_vk}⊂_Λ1$\{{\mathit{v}}_1,\dots ,{\mathit{v}}_k\}\subset {\Lambda }_1$ such that _Λ2${\Lambda }_2$ is, up to similarity, arbitrarily close to the projection of _Λ1${\Lambda }_1$ onto the orthogonal complement of the subspace spanned by {_v1,…,_vk}$\{{\mathit{v}}_1,\dots ,{\mathit{v}}_k\}$. This result extends the main theorem of Sloane et al. (2011) [1] and has applications in communication theory.     

Characterizations of Disjointness preserving operators on vector-valued function spaces

We characterize compact and completely continuous disjointness preserving linear operators on vector-valued continuous functions as follows: a disjointness preserving operator is compact (resp. completely continuous) if and only if

for all

where is continuous and vanishes at infinity in the uniform (resp. strong) operator topology, and is compact (resp.  is uniformly completely continuous).

     

Compact operators in Banach lattices

Disjoint sequence methods from the theory of Riesz spaces are used to study compact operators on Banach lattices. A principal new result of the paper is that each positive map from a Banach latticeE to a Banach latticeF with compact majorant is itself compact provided the norms onE′ andF are order continuous.