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The principal purpose of this announcement is to present an equivalent formulation of the invariant subspace conjecture for bounded linear operators acting on a Hubert space H. Specifically, the conjecture asserts that if B(H) denotes the algebra of bounded linear operators on H and AeB(H\ then A has a nontrivial invariant subspace. We show that the conjecture can be reduced to the study of operators having the property that their invariant subspaces are reducing spaces. In our earlier announcement of this result we called such an operator "completely normal" (cf.[2]); however, since then we have been convinced (by P. R. Halmos) that "reductive" is a more appropriate term. Our basic result is that if dim H > 1, then the invariant subspace conjecture is correct if, and only if, every reductive element of B{H) is normal. Inasmuch as the proof of the result requires an elaborate use of direct integral theory for rings of operators, we have not given proofs to the theorems. The complete proofs are expected to appear in a forthcoming monograph on direct integral theory and its applications. In particular we set H 0 = N(H) and let P 0 denote the projection of AMS 1970 subject classifications. Primary 47A15, 47C15; Secondary 46G10, 46J05.
The principal purpose of this announcement is to present an equivalent formulation of the invariant subspace conjecture for bounded linear operators acting on a Hubert space H. Specifically, the conjecture asserts that if B(H) denotes the algebra of bounded linear operators on H and AeB(H\ then A has a nontrivial invariant subspace. We show that the conjecture can be reduced to the study of operators having the property that their invariant subspaces are reducing spaces. In our earlier announcement of this result we called such an operator "completely normal" (cf.[2]); however, since then we have been convinced (by P. R. Halmos) that "reductive" is a more appropriate term. Our basic result is that if dim H > 1, then the invariant subspace conjecture is correct if, and only if, every reductive element of B{H) is normal. Inasmuch as the proof of the result requires an elaborate use of direct integral theory for rings of operators, we have not given proofs to the theorems. The complete proofs are expected to appear in a forthcoming monograph on direct integral theory and its applications. In particular we set H 0 = N(H) and let P 0 denote the projection of AMS 1970 subject classifications. Primary 47A15, 47C15; Secondary 46G10, 46J05.
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