Rate of decay of s-numbers

Abstract

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