Rate of decay of s-numbers

For an operator T∈B(X,Y), we denote by a_m(T), c_m(T), d_m(T), and t_m(T) its approximation, Gelfand, Kolmogorov, and absolute numbers, respectively. We show that, for any infinite-dimensional Banach spaces X and Y, and any sequence α_m↘0, there exists T∈B(X,Y) for which the inequality 3α_⌈m/6⌉≥a_m(T)≥max{c_m(t),d_m(T)}≥min{c_m(t),d_m(T)}≥t_m(T)≥α_m/9 holds for every m∈N{double-struck}. Similar results are obtained for other s-scales.