The Schur product of two n by n matrices is a pointwise product. More precisely, the Schur product of $m= (m_{ij})$ and $A=(a_{ij})$ is the matrix $(m_{ij}a_{ij})$. Let us fix the matrix m and view the Schur product of matrices with m as a map on the matrix algebra, which we call a Schur multiplier. The boundedness of Schur multipliers (with more general indices) naturally connects to the approximation property of group von Neumann algebras as shown in the work of Haagerup, Lafforgue/de la Salle, Parcet/de la Salle/Ricard, and Parcet etc.

In this talk, I plan to introduce an analogue of Marcinkiewicz-multiplier theory for Schur multipliers on Schatten-p classes. The talk will be based on a recent work with ChianYeong Chuah and Zhenchuan Liu.