Rate of decay of ss-numbers

For an operator T∈B(X,Y)T∈B(X,Y), we denote by am(T)am(T), cm(T)cm(T), dm(T)dm(T), and tm(T)tm(T) its approximation, Gelfand, Kolmogorov, and absolute numbers, respectively. We show that, for any infinite-dimensional Banach spaces XX and YY, and any sequence αm↘0αm↘0, there exists T∈B(X,Y)T∈B(X,Y) for which the inequality 3α⌈m/6⌉⩾am(T)⩾max{cm(t),dm(T)}⩾min{cm(t),dm(T)}⩾tm(T)⩾αm/93α⌈m/6⌉⩾am(T)⩾max{cm(t),dm(T)}⩾min{cm(t),dm(T)}⩾tm(T)⩾αm/9 holds for every m∈Nm∈N. Similar results are obtained for other ss-scales.