Entropy numbers of diagonal operators of logarithmic type. (Q2761595)

In a former paper, \textit{B. Carl} [J. Approximation Theory 32, 135--150 (1981; Zbl 0475.41027)] investigated the degree of compactness of diagonal operators \(D_\sigma\) mapping \(l_p\) into \(l_q\). He characterized those \(\sigma=(\sigma_n)_{n \geq 1}\) for which the sequence \(e_n(D_\sigma)\) of entropy numbers of \(D_\sigma\) belongs to a given Lorentz sequence space. Suppose now that the \(\sigma_n\)'s satisfy \(\sigma_n\preceq n^{-\alpha}(1+\log n)^\delta\) for some \(\alpha\geq 0\) and some \(\delta\in\mathbb R\). Then Carl's results do not lead to similar upper estimates (with some power of \(\log\)) for the entropy numbers of \(D_\sigma\). Those and similar problems are investigated in the paper under review. As an example, let us state one theorem:NEWLINENEWLINETheorem. Let \(1\leq p,q\leq\infty\), \(\alpha>\max(1/q-1/p,0)\) and \(\delta\in\mathbb R\). Then it follows that \(\sigma_n\preceq n^{-\alpha}(1+\log n)^\delta\) iff NEWLINE\[NEWLINE e_n(D_\sigma : l_p\to l_q)\preceq n^{1/q-1/p-\alpha}(1+\log n)^\delta\;. NEWLINE\]NEWLINE The limit case \(\alpha=0\) and \(\delta>0\) is treated as well.