Let M₁ and M₂ be functions on [0,1] such that $M_1\left(t^{1/p}\right)$ and $M_2\left(t^{1/p}\right)$ are Orlicz functions for some $p\in (0,1]$. Assume that $M_2^{-1}(1/t)/M_1^{-1}(1/t)$ is non-decreasing for $t\ge 1$. Let ${({\alpha }_i)}_{i=1}^{\infty }$ be a non-increasing sequence of nonnegative real numbers. Under some conditions on ${({\alpha }_i)}_{i=1}^{\infty }$, sharp two-sided estimates for entropy numbers of diagonal operators $T_{\alpha }:{\ell }_{M_1}\to {\ell }_{M_2}$ generated by ${({\alpha }_i)}_{i=1}^{\infty }$, where ${\ell }_{M_1}$ and ${\ell }_{M_2}$ are Orlicz sequence spaces, are proved. The results generalise some works of Edmunds and Netrusov in [8] and hence a result of Cobos, Kühn and Schonbek in [6].

中文翻译：

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Orlicz 序列空间上对角算子的熵数

令M ₁和M ₂是 [0,1] 上的函数，使得${米}_1\left({吨}^{1/磷}\right)$ 和 ${米}_2\left({吨}^{1/磷}\right)$ 是一些 Orlicz 函数 $磷\in (0,1]$. 假设${米}_2^{-1}(1/吨)/{米}_1^{-1}(1/吨)$ 是不减少的 $吨\ge 1$. 让${({\alpha }_{我})}_{我=1}^{\infty }$是非负实数的非递增序列。在某些条件下${({\alpha }_{我})}_{我=1}^{\infty }$, 对角算子的熵数的锐利两侧估计 ${吨}_{\alpha }：{\ell }_{{米}_1}\to {\ell }_{{米}_2}$ 产生于 ${({\alpha }_{我})}_{我=1}^{\infty }$，哪里 ${\ell }_{{米}_1}$ 和 ${\ell }_{{米}_2}$是 Orlicz 序列空间，得到证明。结果概括了 [8] 中 Edmunds 和 Netrusov 的一些工作，因此是 [6] 中 Cobos、Kühn 和 Schonbek 的结果。