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T. B. Hoover has shown that if A is a reductive operator, then A = A, © A 2, where A t is normal and all the invariant subspaces of A2 are hyperinvariant. A new proof is presented of this result, and several corollaries are derived. Among these is the fact that if A is hyperinvariant and T is polynomially compact and AT = TA, then A*T = TA*. It is also shown that every reductive operator is quasitriangular. Lemma 1. If Ax © A2 is reductive and AXX = XA2, then A\*X = XA*.doi:10.2307/2041072 fatcat:na5zgulrdfddvm5ekq7kcywmva