Entropy numbers of operators acting between vector-valued sequence spaces

Entropy numbers of operators acting between vector-valued sequence spaces are estimated using information about the coordinate mappings. To do this some new ideas of combinatorial type are used. The results are applied to give sharp two-sided estimates of the entropy numbers of some embeddings of Besov spaces. For instance, our main result allows us to give exact two-sided estimates of the entropy numbers of the natural embedding of B-p1,theta 1(w1) (Q) in B-p2 theta 2(w2) (Q), where Q = (0, 1)(d);theta 1,theta 2,p1,p2 epsilon (0, ,infinity], when the condition 1/theta 1 - 1/theta 2 >= 1/p1 - 1/p2 > 0 is satisfied. This work enables us to construct an example showing that the behaviour under real interpolation of entropy numbers can be even worse than in the example of 7.