Characterization of Strong Preserver Operators of Convex Equivalent on the Space of All Real Sequences Tend to Zero

Abstract

‎In this work we consider all bounded linear operators‎

‎$T:\mathfrak{c}_{0}\rightarrow\mathfrak{c}_{0}$‎

‎that preserve convex equivalent relation $\sim_{c}$ on $\mathfrak{c}_{0}$‎

‎and we denote by $\mathcal{P}_{ce}(\mathfrak{c}_{0})$ the set of such operators‎.

‎If $T$ strongly preserves convex‎

‎equivalent‎, ‎we denote them by $\mathcal{P}_{sce}(\mathfrak{c}_{0}).$‎

‎Some interesting properties of $\mathcal{P}_{ce}(\mathfrak{c}_{0})$‎

‎are given‎. ‎For $T\in\mathcal{P}_{ce}(\mathfrak{c}_{0}),$ we show that all‎

‎rows of $T$ belong to $\ell^{1}$ and for any $j\in\mathbb{N},$ we have‎

‎$0\in\mathrm{Im}(T\mathrm{e}_{j}),$ also there are $a,b\in\mathrm{Im}(T\mathrm{e}_{j})$ such‎

‎that $\mathrm{co}(T\mathrm{e}_{j})=[a,b].$ It is shown that any row sums of $T$ belong‎

‎to $[a,b].$ We characterize the elements of $\mathcal{P}_{ce}(\mathfrak{c}_{0}),$ and‎

‎some interesting results of all $T\in\mathcal{P}_{sce}(\mathfrak{c}_{0})$ are given‎, ‎for example‎

‎we prove that $a=0<b$ or $a<0=b.$ Also the elements of $\mathcal{P}_{sce}(\mathfrak{c}_{0})$‎

‎are characterized‎. ‎We obtain the matrix representation of $T\in\mathcal{P}_{sce}(\mathfrak{c}_{0})$‎

‎does not contain any zero row‎. ‎Some relevant examples are given‎.