Dilation to inflations of S(Φ)

Let T be a completely nonunitary contraction with rank (1 - T*T) = 1 on an n-dimensional Hubert space. We prove that (I) if n = 2 and S is an operator which has norm 1, attains its norm and satisfies W(S)⊆ W(T), then S has T as a direct summand, and (2) if ≥ 3 and S is an operator such that S^k dilates to T^k⊕T^k⊕ ... simultaneously for k = 1, 2,..., n - l and W(S)∩∂W(T)≠ θ, then S has T as a direct summand. (Here W(·) denotes the numerical range). These results generalize the corresponding ones for T the n × n nilpotent Jordan block.