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A lattice-theoretic equivalent of the invariant subspace problem

Author: W. E. Longstaff
Journal: Proc. Amer. Math. Soc. 98 (1986), 605-606
MSC: Primary 47A15; Secondary 06B99
MathSciNet review: 861759
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Abstract: Every bounded linear operator on complex infinite-dimensional separable Hilbert space has a proper invariant subspace if and only if for every lattice automorphism $ \phi $ of a certain abstract complete lattice $ P$ (defined by N. Zierler) there exists an element $ a \in P$ different from 0 and 1 such that $ {\phi ^2}(a) \leq a$.

References [Enhancements On Off] (What's this?)

  • [1] P. A. Fillmore and W. E. Longstaff, On isomorphisms of lattices of closed subspaces, Canad. J. Math. 36 (1984), 820-829. MR 762744 (86c:46012)
  • [2] E. A. Nordgren, H. Radjavi, and Peter Rosenthal, A geometric equivalent of the invariant subspace problem, Proc. Amer. Math. Soc. 61 (1976), 66-68. MR 0430822 (55:3827)
  • [3] H. Radjavi and Peter Rosenthal, Invariant subspaces, Springer-Verlag, New York, 1973. MR 0367682 (51:3924)
  • [4] N. Zierler, Axioms for nonrelativistic quantum mechanics, Pacific J. Math. 11 (1961), 1151-1169. MR 0140972 (25:4385)
  • [5] -, On the lattice of closed subspaces of Hilbert space, Pacific J. Math. 19 (1966), 583-586. MR 0202647 (34:2509)

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