Orthogonally Additive Operators on Vector Lattices

Pliev, Marat; Popov, M. M.

doi:10.1007/978-3-031-02104-6_10

Cited by 6 publications

(2 citation statements)

References 43 publications

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“…For the standard information on vector lattices and OAOs, we refer to [3, 20]. Let E be a vector lattice and

x\in E

.…”

Section: Preliminaries and Statements Of Main Resultsmentioning

confidence: 99%

“…It is worth noting that classical operators of nonlinear analysis including Uryson, Hammerstein, and Nemyskii operators are orthogonally additive in appropriate function spaces [14]. The theory of OAOs has been developing by different authors in many directions (see recent survey [20] and the bibliography therein). Some interesting connections of the theory of OAOs with problems of the convex geometry were recently discovered in [26].…”

Section: Orthogonally Additive Operators and Their Relations To Linea...mentioning

confidence: 99%

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Representation theorems for regular operators

Pliev

Popov

2023

Mathematische Nachrichten

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We elaborate, strengthen, and generalize known representation theorems by different authors for regular operators on vector and Banach lattices. Our main result asserts, in particular, that every regular linear operator T acting from a vector lattice E with the principal projection property to a Dedekind complete vector lattice F, which is an ideal of some order continuous Banach lattice G, admits a unique representation , where is the sum of an absolutely order summable family of disjointness preserving operators and is an order narrow (= diffuse) operator. Our main contribution is waiver of the order continuity assumption on T. In proofs, we use new techniques that allow obtaining more general results for a wider class of orthogonally additive operators, which has somewhat different order structure than the linear subspace of linear operators.

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“…For the standard information on vector lattices and OAOs, we refer to [3, 20]. Let E be a vector lattice and

x\in E

.…”

Section: Preliminaries and Statements Of Main Resultsmentioning

confidence: 99%

Section: Orthogonally Additive Operators and Their Relations To Linea...mentioning

confidence: 99%

Representation theorems for regular operators

Pliev

Popov

2023

Mathematische Nachrichten

Self Cite

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$$\varvec{\varepsilon }$$-Shading Operator on Riesz Spaces and Order Continuity of Orthogonally Additive Operators

Mykhaylyuk

Popov

2022

Results Math

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Given a Riesz space E and $$0 < e \in E$$ 0 < e ∈ E , we introduce and study an order continuous orthogonally additive operator which is an $$\varepsilon $$ ε -approximation of the principal lateral band projection $$Q_e$$ Q e (the order discontinuous lattice homomorphism $$Q_e :E \rightarrow E$$ Q e : E → E which assigns to any element $$x \in E$$ x ∈ E the maximal common fragment $$Q_e(x)$$ Q e ( x ) of e and x). This gives a tool for constructing an order continuous orthogonally additive operator with given properties. Using it, we provide the first example of an order discontinuous orthogonally additive operator which is both uniformly-to-order continuous and horizontally-to-order continuous. Another result gives sufficient conditions on Riesz spaces E and F under which such an example does not exist. Our next main result asserts that, if E has the principal projection property and F is a Dedekind complete Riesz space then every order continuous regular orthogonally additive operator $$T :E \rightarrow F$$ T : E → F has order continuous modulus |T|. Finally, we provide an example showing that the latter theorem is not true for $$E = C[0,1]$$ E = C [ 0 , 1 ] and some Dedekind complete F. The above results answer two problems posed in a recent paper by O. Fotiy, I. Krasikova, M. Pliev and the second named author.

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A New Characterization of C-Complete Riesz Spaces

Krasikova,

Mykhaylyuk,

Popov

2024

Results Math

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In the present paper, we expose a deep analogy of some recently discovered objects of the Riesz space theory with classical notions of Analysis such as order ideals, bands, order projections and properties like Dedekind completeness, principal projection property. A Riesz space E is said to be C-complete if every nonempty subset of fragments of an element $$e \in E^+$$ e ∈ E + has a supremum. One of our results asserts that, a Riesz space E is C-complete if and only if every lateral band of E is a projection lateral band. Another result characterizes projective lateral bands in an arbitrary Riesz space and provides an explicit formula for the corresponding lateral band projection, which was previously known for C-complete Riesz spaces. Our characterization of a projective lateral band consists of two conditions, one of which is actually a property of the space, which we call relative C-completeness, and the other one is a property of bands, which we call relative lateral bands. A similar characterization is obtained for the relative C-completeness. We describe the lateral disjoint complement $$A^\dag $$ A † to any subset A of E, show that $$A^{\dag \dag }$$ A † † equals the relative lateral closure of the lateral ideal generated by A and prove that $$A^\dag = A^{\dag \dag \dag }$$ A † = A † † † . Then we obtain some corollaries on the extension of orthogonally additive operators. We also provide many examples confirming the sharpness of our results.

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Orthogonally Additive Operators on Vector Lattices

Cited by 6 publications

References 43 publications

Representation theorems for regular operators

Representation theorems for regular operators

$$\varvec{\varepsilon }$$-Shading Operator on Riesz Spaces and Order Continuity of Orthogonally Additive Operators

A New Characterization of C-Complete Riesz Spaces

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