An Invariant Subspace Theorem

for every rational function ƒ with poles off K. In this note it is shown that any operator for which the spectrum is a spectral set has a nontrivial invariant subspace. In [6] von Neumann introduced the notion of spectral set and showed that if T has II T\\ = 1 then the closed unit disc, D"~, is a spectral set for T. For this reason any operator whose spectrum is a spectral set is called a von Neumann operator. Hence if II 711 = 1 and o(T) = D~ then T is a von Neumann operator. If T or 7* is a subnormal operator then T is a von Neumann operator. Thus the result of this note generalizes the recent result of Scott Brown [1] that every subnormal operator has an invariant subspace, although the proof relies heavily on his techniques. We wish here to thank him for an early manuscript containing his results.