Narrow and ℓ2-strictly singular operators from L p

Mykhaylyuk, Volodymyr; Popov, Mikhail; Randrianantoanina, Beata; Schechtman, Gideon

doi:10.1007/s11856-014-0012-8

Cited by 7 publications

(32 citation statements)

References 9 publications

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“…For the codomain space which is not necessarily a Banach lattice, Mykhaylyuk, Popov, Schechtman, and the present author [16] proved the following result.…”

Section: Flores and Ruizmentioning

confidence: 55%

“…The inclusion embedding J : L p → L r where 1 ≤ r < p < ∞ is an example of a somewhat narrow operator which is not narrow. However, for operators from L p to L p , where 1 ≤ p ≤ 2, these two notions are equivalent as was shown in [16,Theorem 2.2].…”

Section: New Resultsmentioning

confidence: 88%

“…we prove the following result. Before we start the proof of Theorem 2.1 we present a slight weakening of the notion of a narrow operator and a structural result proved in [16], both of which will be useful for our proof. Definition 2.2.…”

Section: New Resultsmentioning

confidence: 99%

“…For example, it is not known whether every narrow operators has an invariant subspace. It also may be interesting to investigate whether semi-embeddings and/or G δ -embeddings may be useful in the setting of spaces of analytic functions, and whether there exists a meaningful analog of sign embeddings and narrow operators in this setting; we note that in [16] narrow operators on L p , 1 < p < 2, were characterized using functions with a controlled growth, instead of sign functions. We refer the reader to [18] where many more open problems and directions of study are discussed.…”

Section: Flores and Ruizmentioning

confidence: 99%

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On sign embeddings and narrow operators on 𝐿₂

Randrianantoanina¹

2017

Problems and Recent Methods in Operator Theory

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The goal of this note is two-fold. First we present a brief overview of "weak" embeddings, with a special emphasis on sign embeddings which were introduced by H. P. Rosenthal in the early 1980s. We also discuss the related notion of narrow operators, which was introduced by A. Plichko and M. Popov in 1990. We give examples of applications of these notions in the geometry of Banach spaces and in other areas of analysis. We also present some open problems.In the second part we prove that Rosenthal's celebrated characterization of narrow operators on L1 is also true for operators on L2. This answers, for p = 2, a question posed by Plichko and Popov in 1990. For 1 < p < 2 the problem remains open.2010 Mathematics Subject Classification. Primary 47B07; secondary 47B38, 46B03, 46E30.

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“…For the codomain space which is not necessarily a Banach lattice, Mykhaylyuk, Popov, Schechtman, and the present author [16] proved the following result.…”

Section: Flores and Ruizmentioning

confidence: 55%

Section: New Resultsmentioning

confidence: 88%

Section: New Resultsmentioning

confidence: 99%

Section: Flores and Ruizmentioning

confidence: 99%

See 2 more Smart Citations

On sign embeddings and narrow operators on 𝐿₂

Randrianantoanina¹

2017

Problems and Recent Methods in Operator Theory

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“…Plichko and Popov were the first [15] who systematically studied this class of operators. Later many authors have studied linear and nonlinear narrow operators in functional spaces and vector lattices [3,4,12,14,17]. In the article [16] the second named author have considered a general lattice-normed space approach to narrow operators.…”

Section: Introductionmentioning

confidence: 99%

Narrow Operators on Lattice-normed Spaces and Vector Measures

Dzadzaeva

Pliev

2016

Trends in Mathematics

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Dedicated to the memory of Adriaan Cornelis Zaanen, the one of the great founder of the theory of vector lattices Abstract. We consider linear narrow operators on lattice-normed spaces. We prove that, under mild assumptions, every finite rank linear operator is strictly narrow (before it was known that such operators are narrow). Then we show that every dominated, order continuous linear operator from a lattice-normed space over atomless vector lattice to an atomic lattice-normed space is order narrow.Mathematics Subject Classification (2010). Primary 46B99; Secondary 46G12.

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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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Narrow and ℓ2-strictly singular operators from L p

Cited by 7 publications

References 9 publications

On sign embeddings and narrow operators on 𝐿₂

On sign embeddings and narrow operators on 𝐿₂

Narrow Operators on Lattice-normed Spaces and Vector Measures

Lateral order on complex vector lattices and narrow operators

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