On the Structure of the Square of a C 0 (1) Operator *

0 While the model theory for contraction operators (cf. [4]) is always a useful tool, it is particularly powerful when dealing with C 0 (1) operators. Recall that an operator T on a Hilbert space H is a C 0 (N)-operator (N = 1, 2. . .) if T ≤ 1, T n → 0 and, T n → 0 (strongly) when n → ∞ and rank(1 − T * T) = N. In particular, a C 0 (1) operator is unitarily equivalent to the compression of the unilateral shift operator S on the Hardy space H 2 to a subspace H 2 ⊖ mH 2 for some inner function m in H ∞. In this note we use the structure theory to determine when the lattices of invariant and hy-perinvariant subspaces differ for the square T 2 of a C 0 (1) operator and the relationship of that to the reducibility of T 2. To accomplish this task we first determine very explicitly the characteristic operator function for T 2 and use the representation obtained to determine when the operator is irreducible. While every operator T in C 0 (1) is irreducible, it does not follow that T 2 is necessarily irreducible, that is, has no reducing subspaces. In particular, we characterize those T in C 0 (1) for which T 2 is irreducible but for which the lattices of invariant and hyperinvariant subspaces for T 2 are distinct. Finally, we provide an example of an operator X on a four dimensional Hilbert space for which the two lattices are distinct but X is irreducible, and show that such an example is not possible on a three dimensional space. This work was prompted by a question to the first author from Ken Dykema (Sect. 2, [2]) concerning hyperinvariant subspaces in von Neumann algebras. He asked whether the lattices of invariant and hyperinvariant subspaces for an irreducible matrix must coincide. He provides an * 2000 AMS Classification: 47A15, 47A45.