Operator synthesis II. Individual synthesis and linear operator equations

The second part of our work on operator synthesis deals with individual operator synthesis of elements in some tensor products, in particular in Varopoulos algebras, and its connection with linear operator equations. Using a developed technique of ``approximate inverse intertwining'' we obtain some generalizations of the Fuglede and the Fuglede-Weiss theorems. Additionally, we give some applications to spectral synthesis in Varopoulos algebras and to partial differential equations.