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.
Let 5C be a complex Hubert Space, J3(3C) the ring of bounded operators on 3C, E an abelian symmetric subring of B(3Z) containing the identity which is closed in the weak operator topology, E\ the commutant of E, and suppose E\ has a cyclic vector £o which we normalize so that |£o| =1. Diximier [l] has shown that E (respect. Ei), as a Banach space, is the dual of the Banach space R (respect. Ri) of all linear forms on E (respect. E\) that are continuous in the ultra-strong topology of E (respect. Ei). In this note we show that every TÇzR is also continuous in the weak operator topology of E, from which it follows that a linear functional T on E is continuous in either the weak, ultraweak, strong, or ultrastrong topologies if and only if it is continuous in all four simultaneously. In the process, we obtain an integral representation for such T, which we later use in a theorem on centrally reducible positive functionals on E%.We denote the maximal ideal space of E by M, and for A, B, • • • £E, we denote the corresponding Gel'fand transforms by a, 6, • • • . Then A->a is an isometric isomorphism from £ onto C(M). Consequently, every bounded linear functional on E has the formwhere v is a complex Borel measure on M uniquely determined by T.Of special interest are functionals of the form A -»(i4 £, £), where | is a vector of 3C. We denote by v$ the measure corresponding to the vector £, and by \x the measure v^. for every AÇzE.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.