Inequalities between entropy and approximation numbers of compact maps
Let H be a Hilbert space, T a compact operator on H having infinite dimensional range, \(a_ n(T)\) the nth approximation number of T, and \(e_ n(T)\) the nth entropy number. It is shown that if N is any positive integer then \[ e_ n(T)\leq 2a_{N+1}(T)\leq 2\sqrt{2}e_{N+2}(T) \] for any integer n for which \[ (n-1)\log 2\geq 2 \log \prod^{N}_{j=1}3a_ j(T)/a_{N+1}(T). \] From these inequalities the author derives others of a similar nature in which bounds for the approximation numbers give corresponding bounds for the entropy numbers.
scientific article; zbMATH DE number 4067657