Triangular projection on 𝑆_{𝑝},0<𝑝<1 and related inequalities

Abstract

In this paper we study properties of the triangular projection P n \mathscr {P}_n on the space of n × n n\times n matrices. The projection P n \mathscr {P}_n annihilates the entries of an n × n n\times n matrix below the main diagonal and leaves the remaining entries unchanged. We estimate the p p -norms of P n \mathscr {P}_n as an operator on the Schatten–von Neumann class S p {\boldsymbol S}_p for 0 > p > 1 0>p>1 . The main result of the paper shows that for p ∈ ( 0 , 1 ) p\in (0,1) , the p p -norms of P n \mathscr {P}_n on S p {\boldsymbol S}_p behave as n → ∞ n\to \infty as n 1 / p − 1 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 S p {\boldsymbol S}_p -quasinorms of the n × n n\times n matrices whose entries above the diagonal are equal to 1 while the entries below the diagonal are equal to 0.