Hankel and Toeplitz–Schur multipliers

Abstract.

We study the problem of characterizing Hankel–Schur multipliers and Toeplitz–Schur multipliers of Schatten–von Neumann class ${\boldsymbol S}_p$ for $0<p<1$. We obtain various sharp necessary conditions and sufficient conditions for a Hankel matrix to be a Schur multiplier of ${\boldsymbol S}_p$. We also give a characterization of the Hankel–Schur multipliers of ${\boldsymbol S}_p$ whos e symbols have lacunary power series. Then the results on Hankel–Schur multipliers are used to obtain a characterization of the Toeplitz–Schur multipliers of ${\boldsymbol S}_p$. Finally, we return to Hankel–Schur multipliers and obtain new results in the case when the symbol of the Hankel matrix is a complex measure on the unit circle.