Maximal Contractive Tuples

Maximality of a contractive tuple of operators is considered. A characterization for a contractive tuple to be maximal is obtained. The notion of maximality for a submodule of the Drury-Arveson module on the -dimensional unit ball is defined. For , it is shown that every submodule of the Hardy module over the unit disc is maximal. But for we prove that any homogeneous submodule or submodule generated by polynomials is not maximal. A characterization of maximal submodules is obtained.