Triangular projection on ${\mathit{S}}_p,0<p<1$, as $\mathit{p}$ approaches 1

This is a continuation of our recent paper. We continue studying properties of the triangular projection ${\mathcal{P}}_n$ on the space of $n\times n$ matrices. We establish sharp estimates for the $p$-norms of ${\mathcal{P}}_n$ as an operator on the Schatten–von Neumann class ${\mathit{S}}_p$ for $0<p<1$. Our estimates are uniform in $n$ and $p$ as soon as $p$ is separated away from 0. 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 $n\to \mathrm{\infty }$ and $p\to 1$ as $n^{1\mathrm{/}p-1}\min \{(1-p{)}^{-1},\log n\}$.