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 ${\alpha }_m\searrow 0$, there exists $T\in B(X,Y)$ for which the inequality $$3{\alpha }_{\lceil m/6\rceil }\ge a_m(T)\ge \max \{c_m(t),d_m(T)\}\ge \min \{c_m(t),d_m(T)\}\ge t_m(T)\ge {\alpha }_m/9$$ holds for every $m\in \mathbb{N}$. Similar results are obtained for other $s$-scales.

References in corpus (1)

• Approximation schemes satisfying Shapiro's Theorem