This is an announcement for the paper “Certain geometric structure of Λ-sequence spaces” by Atanu Mannahttps://arxiv.org/find/math/1/au:+Manna_A/0/1/0/all/0/1.

Abstract: The $Lambda$-sequence spaces $\Lambda_p$ for $1<p\leq\infty$ and its generalization $\Lambda_{\hat{p}}$ for $1<\hat{p}<\infty, \hat{p}=(p_n)$ is introduced. The James constants and strong $n$-th James constants of $\Lambda_p$ for $1<p\leq\infty$ is determined. It is proved that generalized $\Lambda$-sequence space $\Lambda_{\hat{p}}$ is embedded isometrically in the Nakano sequence space $\ell_{\hat{p}}(R_{n+1})$ of finite dimensional Euclidean space $R_{n+1}$. Hence it follows that sequence spaces $\Lambda_{p}$ and $\Lambda_{\hat{p}}$ possesses the uniform Opial property, property ($\beta$) 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 $\Lambda_2^{(2)}$ is being carried out.

The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.01519