Triangular projection on ${\mathit{S}}_p,0<p<1,$ and related inequalities

In this paper we study properties of the triangular projection ${\mathcal{P}}_n$ on the space of $n\times n$ matrices. The projection ${\mathcal{P}}_n$ annihilates the entries of an $n\times n$ matrix below the main diagonal and leaves the remaining entries unchanged. We estimate the $p$-norms of ${\mathcal{P}}_n$ as an operator on the Schatten–von Neumann class ${\mathit{S}}_p$ for $0<p<1$. The main result of the paper shows that for $p\in (0,1)$, the $p$-norms of ${\mathcal{P}}_n$ on ${\mathit{S}}_p$ behave as $n\to \mathrm{\infty }$ as $n^{1\mathrm{/}p-1}$. This solves a problem posed by B.S. Kashin. Among other results of this paper we mention the result that describes the behaviour of the ${\mathit{S}}_p$-quasinorms of the $n\times n$ matrices whose entries above the diagonal are equal to 1 while the entries below the diagonal are equal to 0.