Some relations between entropy and approximation numbers (Q1585564)

The author relates the decay of the entropy numbers \(\varepsilon_k(T)\) of operators \(T\) in Hilbert spaces to the decay of their singular numbers \(s_n(T)\). He discusses various examples of exponential decay which is of different order for the singular and entropy numbers. He does not seem to be aware of the general formula \[ e_{k+ 1}(T)\sim \sup_{n\in\mathbb{N}} 2^{-k/n}(s_1(T)\cdots s_n(T))^{1/n} \] shown by \textit{Y. Gordon}, \textit{H. König} and \textit{C. Schütt} in [J. Approximation Theory 49, 219-239 (1987; Zbl 0647.47035)] from which his results can be deduced to a large extent.