Rate of decay of s-numbers

TL;DR

This paper investigates the decay rates of various s-numbers of operators between infinite-dimensional Banach spaces, demonstrating the existence of operators with prescribed decay behaviors for all sequences tending to zero.

Contribution

It establishes the existence of operators with specific decay inequalities for approximation, Gelfand, Kolmogorov, and absolute numbers across all sequences tending to zero.

Findings

Constructs operators with prescribed decay rates for s-numbers.

Shows inequalities linking different s-numbers for these operators.

Applies results to various s-scale sequences.

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 $$ 3 \alpha_{\lceil m/6 \rceil} \geq a_m(T) \geq \max\{c_m(t), d_m(T)\} \geq \min\{c_m(t), d_m(T)\} \geq t_m(T) \geq \alpha_m/9 $$ holds for every $m \in \N$. Similar results are obtained for other $s$-scales.