Rate of decay of the Bernstein numbers (Q2848853)

This paper gives the background material and proof to the result: ``If a Banach space \(X\) contains uniformly complemented \(\ell^n+2\)'s, then there exists a universal constant \(b= b(X)> 0\) such that for each Banach space \(Y\) and any sequence \(d_n\uparrow 0\), there is a bounded linear operator \(T: X\to Y\) with the Bernstein numbers \(b_n(T)\) of \(T\) satisfying \(b^{-1}d_n\leq b_n(T)\leq bd_n\) for all \(n\).''