Consider the set of all Toeplitz-Schur multipliers sending every upper triangular matrix from the trace class into a matrix with absolutely summable entries. We show that this set admits a description completely analogous to that of the set of all Fourier multipliers from H₁ into ℓ₁. We characterize the set of all Schur multipliers sending matrices representing bounded operators on ℓ₂ into matrices with absolutely summable entries. Next, we present a result (due to G. Pisier) that the upper triangular parts of such Schur multipliers are precisely the Schur multipliers sending upper triangular parts of matrices representing bounded linear operators on ℓ₂ into matrices with absolutely summable entries. Finally, we complement solutions of Mazur's Problems 8 and 88 in the Scottish Book concerning Hankel matrices.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm175-2-5,
author = {Aleksander Pe\l czy\'nski and Fyodor Sukochev},
title = {Some remarks on Toeplitz multipliers and Hankel matrices},
journal = {Studia Mathematica},
volume = {173},
year = {2006},
pages = {175-204},
zbl = {1105.47025},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm175-2-5}
}
Aleksander Pełczyński; Fyodor Sukochev. Some remarks on Toeplitz multipliers and Hankel matrices. Studia Mathematica, Tome 173 (2006) pp. 175-204. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm175-2-5/