Advances in Operator Theory

Certain geometric structures of $\Lambda$-sequence spaces

Atanu Manna

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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.

Article information

Adv. Oper. Theory Volume 3, Number 2 (2018), 433-450.

Received: 15 May 2017
Accepted: 27 November 2017
First available in Project Euclid: 15 December 2017

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Digital Object Identifier

Primary: 46B20: Geometry and structure of normed linear spaces
Secondary: 26D15: Inequalities for sums, series and integrals 46A45: Sequence spaces (including Köthe sequence spaces) [See also 46B45] 46B45: Banach sequence spaces [See also 46A45] 40G05: Cesàro, Euler, Nörlund and Hausdorff methods

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


Manna, Atanu. Certain geometric structures of $\Lambda$-sequence spaces. Adv. Oper. Theory 3 (2018), no. 2, 433--450. doi:10.15352/AOT.1705-1164.

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