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Spring 2018 Certain geometric structures of $\Lambda$-sequence spaces
Atanu Manna
Adv. Oper. Theory 3(2): 433-450 (Spring 2018). DOI: 10.15352/AOT.1705-1164

Abstract

The $\Lambda$-sequence spaces $\Lambda_p$ for $1 \lt p\leq\infty$ and their generalized forms $\Lambda_{\hat{p}}$ for $1 \lt \hat{p} \lt \infty$, $\hat{p}=(p_n)$, $n\in \mathbb{N}_0$ are introduced. The James constants and strong $n$-th James constants of $\Lambda_p$ for $1 \lt p \leq \infty$ are determined. It is proved that the generalized $\Lambda$-sequence space $\Lambda_{\hat{p}}$ is a closed subspace of the Nakano sequence space $l_{\hat{p}}(\mathbb{R}^{n+1})$ of finite dimensional Euclidean space $\mathbb{R}^{n+1}$, $n\in \mathbb{N}_0$. Hence it follows that sequence spaces $\Lambda_p$ and $\Lambda_{\hat{p}}$ possess the uniform Opial property, ($\beta$)-property of Rolewicz, and weak uniform normal structure. Moreover, it is established that $\Lambda_{\hat{p}}$ possesses the coordinate wise uniform Kadec–Klee property. Further, necessary and sufficient conditions for element $x\in S(\Lambda_{\hat{p}})$ to be an extreme point of $B(\Lambda_{\hat{p}})$ are derived. Finally, estimation of von Neumann-Jordan and James constants of two dimensional $\Lambda$-sequence space $\Lambda_2^{(2)}$ are carried out. Upper bound for the Hausdorff matrix operator norm on the non-absolute type $\Lambda$-sequence spaces is also obtained.

Citation

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Atanu Manna. "Certain geometric structures of $\Lambda$-sequence spaces." Adv. Oper. Theory 3 (2) 433 - 450, Spring 2018. https://doi.org/10.15352/AOT.1705-1164

Information

Received: 15 May 2017; Accepted: 27 November 2017; Published: Spring 2018
First available in Project Euclid: 15 December 2017

zbMATH: 06848511
MathSciNet: MR3738223
Digital Object Identifier: 10.15352/AOT.1705-1164

Subjects:
Primary: 46B20
Secondary: 26D15 , 40G05 , 46A45 , 46B45

Keywords: Cesàro sequence space , extreme point , Hausdorff method , James constant , Kadec-Klee property , Nakano sequence space , ‎von ‎‎Neumann-Jordan constant

Rights: Copyright © 2018 Tusi Mathematical Research Group

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Vol.3 • No. 2 • Spring 2018
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