# Projecting onto Helson matrices in Schatten classes

## TL;DR

This paper investigates the boundedness of projections onto Helson matrices within Schatten classes, revealing that such projections are unbounded for all Schatten classes except the Hilbert--Schmidt class.

## Contribution

It proves that the orthogonal projection onto Helson matrices is unbounded in Schatten classes $\\mathcal{S}_q$ for all $q
eq 2$, extending understanding of operator bounds in these classes.

## Key findings

- Projection onto Helson matrices is unbounded in $\mathcal{S}_q$ for $q \neq 2$
- The result applies to a broad class of natural projections
- Provides new insights into the structure of Helson matrices in Schatten classes

## Abstract

A Helson matrix is an infinite matrix $A = (a_{m,n})_{m,n\geq1}$ 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 $1 \leq q \neq 2 < \infty$. 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 $1 \leq q \neq 2 < \infty$. Two additional results are also presented.