On One Extremal Problem for a Seminorm on the Space $l_1$ with Weight

Abstract

Let $α=\{α_j\}_{j∈N}$ be a nondecreasing sequence of positive numbers and let $l_{1,α}$ be the space of real sequences $ξ=\{ξ_j\}_{j∈N}$ for which $∥ξ∥_{1,α} := ∑^{∞}_{j=1}α_j|ξ_j| < +∞$. We associate every sequence $ξ$ from $l_{1,α}$ with a sequence $ξ^∗ = \{|ξ_{φ(j)}|\}_{j∈N}$, where $ϕ(·)$ is a permutation of the natural series such that $|ξ_{φ(j)}| ⩾ |ξ_{φ(j+1)}|,\; j ∈ ℕ$. If $p$ is a bounded seminorm on $l_{1,α}$ and $\omega _m :\; = \left\{ {\underbrace {1, \ldots ,1}_m,\;0,\;0,\; \ldots } \right\}$, then $$\mathop {\sup }\limits_{\xi \ne 0,\;\xi \ne 1_{1,\alpha } } \frac{{p\left( {\xi *} \right)}}{{\left\| \xi \right\|_{1,\alpha } }} = \mathop {\sup }\limits_{m \in \mathbb{N}} \frac{{p\left( {\omega _m } \right)}}{{\sum {_{s = 1}^m } \alpha _s }}.$$ Using this equality, we obtain several known statements.