Let $l_0$ be the group (with respect to the coordinate-wise addition) of all sequences of real numbers $x={\left(x_k\right)}_{k=1}^{\infty }$ that are eventually zero, equipped with the quasi-norm ${\Vert x\Vert }_0=\mathrm{card}\{suppx\}$. A description of orbits of elements in the pair $(l_0,l_1)$ is given, which complements (in the sequence space setting) the classical Calderón–Mityagin theorem on a description of orbits of elements in the pair $(l_1,l_{\infty })$. As a consequence, we obtain that the pair $(l_0,l_1)$ is $\mathcal{K}$-monotone.

中文翻译：

---

重排不变组类别的Calderón–Mityagin定理的一个版本

让 ${升}_0$ 是所有实数序列的组（相对于坐标相加） $X={（X_{ķ}）}_{ķ=\mathrm{1个}}^{\infty }$ 最终为零，并配备了准范数 ${\Vert X\Vert }_0=\mathrm{卡}\{支持X\}$。对中元素轨道的描述$（{升}_0，{升}_{\mathrm{1个}}）$ 给出了对（在序列空间设置中）经典Calderón–Mityagin定理的补充，该定理基于该对中元素的轨道描述 $（{升}_{\mathrm{1个}}，{升}_{\infty }）$。结果，我们得到了$（{升}_0，{升}_{\mathrm{1个}}）$ 是 $\mathcal{ķ}$-单调。