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Maximal contractive tuples

Maximality of a contractive tuple of operators is considered. Characterization of a contractive tuple to be maximal is obtained. Notion of maximality of a submodule of Drury-Arveson module on the $d$-dimensional unit ball $\mathbb{B}_d$ is defined. For $d=1$, it is shown that every submodule of the Hardy module over the unit disc is maximal. But for $d\ge 2$ we prove that any homogeneous submodule or submodule generated by polynomials is not maximal. A characterization of a submodule to be maximal is obtained.