Schur multipliers of the Schatten–von Neumann class $S_p$ for $0<p<1$.

I am going to speak about my joint results with A. B. Aleksandrov. We will discuss various conditions for the membership in the space of Schur multipliers of the Schatten–von Neumann class $S_p$. In particular, we are going to discuss the problem by G. Pisier of whether a Schur multiplier of $S_p$ for $1<p<2$ must be completely bounded. We were able to prove that this is the case for $0<p<1$.