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

TL;DR

This paper investigates the behavior of the triangular projection operator on Schatten classes for 0<p<1, establishing how its p-norms scale with matrix size and solving a problem posed by Kashin.

Contribution

It provides the first precise asymptotic estimates of the p-norms of the triangular projection on $oldsymbol{S}_p$ for 0<p<1, addressing a previously open problem.

Findings

The p-norms of the projection grow as n^{1/p-1} for large n.

The paper characterizes the $oldsymbol{S}_p$-quasinorms of matrices with specific entry patterns.

It confirms the asymptotic behavior of the projection norms as n approaches infinity.

Abstract

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 $\boldsymbol{S}_p$ 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 $\boldsymbol{S}_p$ behave as $n\to\infty$ as $n^{1/p-1}$. 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 $\boldsymbol{S}_p$-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.