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 and solve some problems posed in [ Open Problems, Proc. Fourth Conf. Operator Theory (Timişoara/Herculane 1979), Univ. Timişoara and Nat. Inst. Sci. Tech. Creation, Timişoara (1980), 335–342.], [Gary Weiss, An extension of the Fuglede-Putnam theorem modulo the Hilbert-Schmidt class to operators of the form ΣM n XN n, Trans. Amer. Math. Soc. 278 (1983), no. 1, 1–20.], [ Gary Weiss, The Fuglede commutativity theorem modulo the Hilbert-Schmidt class and generating functions for matrix operators. II, J. Oper. Th. 5 (1981), no. 1, 3–16. ]. Additionally, we give some applications to spectral synthesis in Varopoulos algebras and to partial differential equations.