Rate of decay of \(s\)-numbers (Q630497)

Let \(X\) and \(Y\) be infinite-dimensional Banach spaces and \(T\in B(X,Y)\) a linear operator. Define its approximation numbers \(a_m(T)\), Kolmogorov numbers \(d_m(T)\) and Gelfand numbers \(c_m(T)\). Then the author shows that for any infinite-dimensional Banach spaces \(X\) and \(Y\) and any sequence \(a_m\) converging to 0, there exists a \(T\in B(X,Y)\) for which the inequality \[ 3\alpha_{[m/6]}\geq a_m(T)\geq \max\{c_m(t),d_m(T)\}\geq \min\{c_m(t),d_m(T)\}\geq t_m(T)\geq \alpha_m/9 \] holds for each \(m\in N\). Similar results are obtained for other \(s\)-scales.