  We study the problem of characterizing Hankel-Schur multipliers and
Toeplitz-Schur multipliers of Schatten-von Neumann class $\bS_p$ for $0<p<1$.
We obtain various sharp necessary conditions and sufficient conditions for a
Hankel matrix to be a Schur multiplier of $\bS_p$. We also give a
characterization of the Hankel-Schur multipliers of $\bS_p$ whose symbols have
lacunary power series. Then the results on Hankel-Schur multipliers are used to
obtain a characterization of the Toeplitz-Schur multipliers of $\bS_p$.
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.
