Entropy numbers and interpolation

This paper settles a long-standing question by showing that in certain circumstances the entropy numbers of a map do not behave well under real interpolation. To do this, lemmas of combinatorial type are established and used to obtain lower bounds for the entropy numbers of a particular diagonal map acting between Lorentz sequence spaces. These lower bounds contradict the estimates from above that would be obtained if the behaviour of entropy numbers under real interpolation was as good as conjectured. The paper also provides sharp two-sided estimates of the entropy number e _n (T) of diagonal operators \({T:l_{p}\rightarrow l_{q}, T(\left( a_{k}\right)_{k=1}^{\infty}) = (( \lambda_{k}a_{k}) _{k=1}^{\infty}) ,}\) where 0 < p < q ≤ ∞ and \({\{\lambda _{i}\}_{i=1}^{\infty}}\) is a non-increasing sequence of non-negative numbers with λ _i  = λ _n for all i ≤ n.