Projections onto the set of Hankel matrices

The article is devoted to the estimates from below of the norms of projections onto the set of $(n\times n)$ Hankel matrices. Let $B_N$ be the set of operators $T\sim\{t_{jk}\}_{j, k\geqslant0}$ on $l^2$ such that $t_{jk}=0$ for $k+j>N$ and $\mathrm{Hank}_N$ be the subspace of $B_N$ consisting of those operators $T$ for which $t_{jk}=c_{j+k}$ (Hankel matrices). The numbers $\alpha_N$ are defined as the infimum of the norms of projections from $B_N$ onto$\mathrm{Hank}_N$. The main result of the article claims that $c_1\left(\frac{\log N}{\log\log N}\right)^{1/2}\leqslant\alpha_N\leqslant c_2(\log N)^{1/2}$.