Points of narrowness and uniformly narrow operators
It is known that the sum of every two narrow operators on $L_1$ is narrow, however the same is false for $L_p$ with $1 < p < \infty$. The present paper continues numerous investigations of the kind. Firstly, we study narrowness of a linear and orthogonally additive operators on Kothe function...
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Vasyl Stefanyk Precarpathian National University
2017-06-01
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doaj-d78537cb615e4a7c9619382c96e130b52020-11-25T03:06:43ZengVasyl Stefanyk Precarpathian National UniversityKarpatsʹkì Matematičnì Publìkacìï2075-98272313-02102017-06-0191374710.15330/cmp.9.1.37-471445Points of narrowness and uniformly narrow operatorsA.I. Gumenchuk0I.V. Krasikova1M.M. Popov2Chernivtsi Medical College, 60 Geroiv Maidanu str., 58001, Chernivtsi, UkraineZaporizhzhya National University, 66 Zukovs'koho str., 69600, Zaporizhzhya, UkraineVasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, UkraineIt is known that the sum of every two narrow operators on $L_1$ is narrow, however the same is false for $L_p$ with $1 < p < \infty$. The present paper continues numerous investigations of the kind. Firstly, we study narrowness of a linear and orthogonally additive operators on Kothe function spaces and Riesz spaces at a fixed point. Theorem 1 asserts that, for every Kothe Banach space $E$ on a finite atomless measure space there exist continuous linear operators $S,T: E \to E$ which are narrow at some fixed point but the sum $S+T$ is not narrow at the same point. Secondly, we introduce and study uniformly narrow pairs of operators $S,T: E \to X$, that is, for every $e \in E$ and every $\varepsilon > 0$ there exists a decomposition $e = e' + e''$ to disjoint elements such that $\|S(e') - S(e'')\| < \varepsilon$ and $\|T(e') - T(e'')\| < \varepsilon$. The standard tool in the literature to prove the narrowness of the sum of two narrow operators $S+T$ is to show that the pair $S,T$ is uniformly narrow. We study the question of whether every pair of narrow operators with narrow sum is uniformly narrow. Having no counterexample, we prove several theorems showing that the answer is affirmative for some partial cases.https://journals.pnu.edu.ua/index.php/cmp/article/view/1445narrow operatororthogonally additive operatorkothe banach space |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
A.I. Gumenchuk I.V. Krasikova M.M. Popov |
spellingShingle |
A.I. Gumenchuk I.V. Krasikova M.M. Popov Points of narrowness and uniformly narrow operators Karpatsʹkì Matematičnì Publìkacìï narrow operator orthogonally additive operator kothe banach space |
author_facet |
A.I. Gumenchuk I.V. Krasikova M.M. Popov |
author_sort |
A.I. Gumenchuk |
title |
Points of narrowness and uniformly narrow operators |
title_short |
Points of narrowness and uniformly narrow operators |
title_full |
Points of narrowness and uniformly narrow operators |
title_fullStr |
Points of narrowness and uniformly narrow operators |
title_full_unstemmed |
Points of narrowness and uniformly narrow operators |
title_sort |
points of narrowness and uniformly narrow operators |
publisher |
Vasyl Stefanyk Precarpathian National University |
series |
Karpatsʹkì Matematičnì Publìkacìï |
issn |
2075-9827 2313-0210 |
publishDate |
2017-06-01 |
description |
It is known that the sum of every two narrow operators on $L_1$ is narrow, however the same is false for $L_p$ with $1 < p < \infty$. The present paper continues numerous investigations of the kind. Firstly, we study narrowness of a linear and orthogonally additive operators on Kothe function spaces and Riesz spaces at a fixed point. Theorem 1 asserts that, for every Kothe Banach space $E$ on a finite atomless measure space there exist continuous linear operators $S,T: E \to E$ which are narrow at some fixed point but the sum $S+T$ is not narrow at the same point. Secondly, we introduce and study uniformly narrow pairs of operators $S,T: E \to X$, that is, for every $e \in E$ and every $\varepsilon > 0$ there exists a decomposition $e = e' + e''$ to disjoint elements such that $\|S(e') - S(e'')\| < \varepsilon$ and $\|T(e') - T(e'')\| < \varepsilon$. The standard tool in the literature to prove the narrowness of the sum of two narrow operators $S+T$ is to show that the pair $S,T$ is uniformly narrow. We study the question of whether every pair of narrow operators with narrow sum is uniformly narrow. Having no counterexample, we prove several theorems showing that the answer is affirmative for some partial cases. |
topic |
narrow operator orthogonally additive operator kothe banach space |
url |
https://journals.pnu.edu.ua/index.php/cmp/article/view/1445 |
work_keys_str_mv |
AT aigumenchuk pointsofnarrownessanduniformlynarrowoperators AT ivkrasikova pointsofnarrownessanduniformlynarrowoperators AT mmpopov pointsofnarrownessanduniformlynarrowoperators |
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