Rate of Decay of the Bernstein Numbers

We show that if a Banach space $X$ contains uniformly complemented $\ell_2^n$'s then there exists a universal constant $b=b(X)>0$ such that for each Banach space $Y$, and any sequence $d_n\downarrow 0$ there is a bounded linear operator $T:X\to Y$ with the Bernstein numbers $b_n(T)$ of $T$ satisfying $b^{-1}d_n\le b_n(T)\le bd_n$ for all $n$.