The present article is a survey of known results on Schur and operator multipliers. It starts with the classical description of Schur multipliers due to Grothendieck, followed by a discussion of measurable Schur multipliers and a generalisation of Grothendieck's Theorem due to Peller. Thereafter, a non-commutative version of Schur multipliers, called operator multipliers and introduced by Kissin and Schulman, is discussed, and a characterisation extending the description in the commutative case...

Relations between spectral synthesis in the Fourier algebra A(G) of a compact group G and the concept of operator synthesis due to Arveson have been studied in the literature. For an A(G)-submodule X of VN(G), X-synthesis in A(G) has been introduced by E. Kaniuth and A. Lau and studied recently by the present authors. To any such X we associate a $V^{\infty }\left(G\right)$-submodule X̂ of ℬ(L²(G)) (where $V^{\infty }\left(G\right)$ is the weak-* Haagerup tensor product $L^{\infty }\left(G\right){\otimes }_{w*h}{L}^{\infty }\left(G\right)$), define the concept of X̂-operator synthesis and prove that a closed set E...