Projecting onto Helson matrices in Schatten classes
Volume 256 / 2021
Studia Mathematica 256 (2021), 109-119
MSC: Primary 47B35; Secondary 47B10.
DOI: 10.4064/sm190901-24-3
Published online: 15 June 2020
Abstract
A Helson matrix is an infinite matrix $A = (a_{m,n})_{m,n\geq 1}$ 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 \leq q \neq 2 \lt \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 \lt \infty $. Two additional results are also presented.
