Polar decomposition of order bounded disjointness preserving operators

Boulabiar, Karim; Buskes, Gerard

doi:10.1090/s0002-9939-03-07007-2

Cited by 10 publications

(2 citation statements)

References 13 publications

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“…We observe that there is a difference between the real and complex narrowness of linear operators and an investigation of complex narrow operators is the single challenging problem. It is worth noting that different classes of linear operators on complex vector lattices were studied by numbers of authors (see [5,9,12,23]).…”

Section: Introductionmentioning

confidence: 99%

Lateral order on complex vector lattices and narrow operators

Dzhusoeva

Huang

Pliev

et al. 2023

Mathematische Nachrichten

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In this paper, we continue investigation of the lateral order on vector lattices started in [25]. We consider the complexification 𝐸 ℂ of a real vector lattice 𝐸 and introduce the lateral order on 𝐸 ℂ . Our first main result asserts that the set of all fragments 𝔉 𝑣 of an element 𝑣 ∈ 𝐸 ℂ of the complexification of an uniformly complete vector lattice 𝐸 is a Boolean algebra. Then, we study narrow operators defined on the complexification 𝐸 ℂ of a vector lattice 𝐸, extending the results of articles [22,27,28] to the setting of operators defined on complex vector lattices.We prove that every order-to-norm continuous linear operator  ∶ 𝐸 ℂ → 𝑋 from the complexification 𝐸 ℂ of an atomless Dedekind complete vector lattice 𝐸 to a finite-dimensional Banach space 𝑋 is strictly narrow. Then, we prove that every 𝐶-compact order-to-norm continuous linear operator  from 𝐸 ℂ to a Banach space 𝑋 is narrow. We also show that every regular order-no-norm continuous linear operator from 𝐸 ℂ to a complex Banach lattice (𝓁 𝑝 () ℂ is narrow. Finally, in the last part of the paper we investigate narrow operators taking values in symmetric ideals of compact operators.

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Section: Introductionmentioning

confidence: 99%

Lateral order on complex vector lattices and narrow operators

Dzhusoeva

Huang

Pliev

et al. 2023

Mathematische Nachrichten

View full text Add to dashboard Cite

show abstract

“…On the other hand, invertible disjointness preserving operators occupied a prominent role in a vast literature, such as [7,20,33,35] and mainly the remarkable memoir [3] by Abramovich and Kitover. One of the external reasons for the continuing interest in disjointness preserving operators is the fact that precisely the order bounded disjointness preserving operators allow multiplicative representations as weighted composition operators and, more generally, polar decompositions [2,19,28,39]. They thus found applications in the theory of singular and integral equation, dynamical system, and differential equations with delayed time [38,46,51].…”

Section: Introductionmentioning

confidence: 99%