Some remarks on Toeplitz multipliers and Hankel matrices}

Aleksander Pe/lczy/nski¹ ; Fyodor Sukochev²

¹Institute of Mathematics Polish Academy of Sciences /Sniadeckich 8 00-956 Warszawa, Poland
²School of Informatics and Engineering Flinders University of South Australia 5042 Bedford Park, Australia

Studia Mathematica, Tome 175 (2006) no. 2, pp. 175-204

Résumé

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 $H_1$ into $\ell_1$. We characterize the set of all Schur multipliers sending matrices representing bounded operators on $\ell_2$ 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 $\ell_2$ into matrices with absolutely summable entries. Finally, we complement solutions of Mazur's Problems 8 and 88 in the Scottish Book concerning Hankel matrices.

Détail

DOI : 10.4064/sm175-2-5

Keywords: consider set toeplitz schur multipliers sending every upper triangular matrix trace class matrix absolutely summable entries set admits description completely analogous set fourier multipliers nbsp nbsp ell characterize set schur multipliers sending matrices representing bounded operators nbsp ell matrices absolutely summable entries present result due nbsp pisier upper triangular parts schur multipliers precisely schur multipliers sending upper triangular parts matrices representing bounded linear operators nbsp ell matrices absolutely summable entries finally complement solutions mazurs problems scottish book concerning hankel matrices