The sequence of entropy numbers quantifies the degree of compactness of a linear operator acting between quasi-Banach spaces. We determine the asymptotic behavior of entropy numbers in the case of natural embeddings between finite-dimensional Lorentz spaces $\ell_{p,q}^n$ in all regimes; our results are sharp up to constants. This generalizes classical results obtained by Sch\"utt (in the case of Banach spaces) and Edmunds and Triebel, K\"uhn, as well as Gu\'edon and Litvak (in the case of quasi-Banach spaces) for entropy numbers of identities between Lebesgue sequence spaces $\ell_p^n$. We employ techniques such as interpolation, volume comparison as well as techniques from sparse approximation and combinatorial arguments. Further, we characterize entropy numbers of embeddings between symmetric quasi-Banach spaces satisfying a lattice property in terms of best $s$-term approximation numbers.

翻译：熵数列量化了拟巴拿赫空间之间线性算子的紧致程度。我们确定了有限维洛伦兹空间$\ell_{p,q}^n$自然嵌入在全部参数范围内的熵数渐近行为；所得结果在常数意义下是精确的。这推广了Schütt（巴拿赫空间情形）、Edmunds与Triebel、Kühn，以及Guédon与Litvak（拟巴拿赫空间情形）关于勒贝格序列空间$\ell_p^n$恒等嵌入熵数的经典结果。我们采用了插值法、体积比较法，以及稀疏逼近技术与组合论证等技巧。进一步，我们利用最优$s$项逼近数刻画了满足格性质的对称拟巴拿赫空间之间嵌入的熵数。