Projecting onto Helson matrices in Schatten classes

Abstract

A Helson matrix is an infinite matrix $A=(a_{m,n}{)}_{m,n\ge 1}$ such that the entry $a_{m,n}$ depends only on the product $mn$. We demonstrate that the orthogonal projection from the Hilbert--Schmidt class ${\mathcal{S}}_2$ onto the subspace of Hilbert--Schmidt Helson matrices does not extend to a bounded operator on the Schatten class ${\mathcal{S}}_q$ for $1\le q\ne 2<\infty$. In fact, we prove a more general result showing that a large class of natural projections onto Helson matrices are unbounded in the ${\mathcal{S}}_q$-norm for $1\le q\ne 2<\infty$. Two additional results are also presented.

This paper has been has been accepted for publication in Studia Math