Entropy numbers of diagonal operators on Orlicz sequence spaces

TL;DR

This paper derives precise estimates for the entropy numbers of diagonal operators between Orlicz sequence spaces, extending previous results and providing a deeper understanding of their compactness properties.

Contribution

It provides sharp two-sided bounds for entropy numbers of diagonal operators on Orlicz sequence spaces under new conditions, generalizing prior work by Edmunds, Netrusov, Cobos, K"uhn, and Schonbek.

Findings

Established sharp two-sided estimates for entropy numbers.

Extended previous results to broader classes of Orlicz spaces.

Generalized known bounds for specific diagonal operators.

Abstract

Let $M_1$ and $M_2$ be functions on $[0,1]$ such that $M_1(t^{1/p})$ and $M_2(t^{1/p})$ 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 $t \geq 1.$ Let $(\alpha_i)_{i=1}^\infty$ be a non-increasing sequence of non-negative real numbers. Under some conditions on $(\alpha_i)_{i=1}^\infty,$ sharp two-sided estimates for entropy numbers of diagonal operators $T_\alpha :\ell_{M_1} \rightarrow \ell_{M_2}$ 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 and hence a result of Cobos, K\"{u}hn and Schonbek.