  We introduce the defect sequence for a contractive tuple of Hilbert space
operators and investigate its properties. The defect sequence is a sequence of
numbers, called defect dimensions associated with a contractive tuple. We show
that there are upper bounds for the defect dimensions. The tuples for which
these upper bounds are obtained, are called maximal contractive tuples. The
upper bounds are different in the non-commutative and in the com- mutative
case. We show that the creation operators on the full Fock space and the co
ordinate multipliers on the Drury-Arveson space are maximal. We also study pure
tuples and see how the defect dimensions play a role in their irreducibility.
