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Small Transitive Lattices

D. W. Hadwin, W. E. Longstaff, Peter Rosenthal

1983
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Proceedings of the American Mathematical Society
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Partial results are obtained on the problem of determining the smallest lattice of subspaces of a Hubert space with the property that the only operators leaving all the subspaces invariant are the multiples of the identity. Halmos [3] initiated the study of transitive lattices of subspaces of Hubert space, i.e., subspace lattices with the property that the only (bounded) operators leaving all the subspaces in the lattice invariant are scalar multiples of the identity operator. By definition,
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... . By definition, every subspace lattice contains the two trivial elements, {0} and the entire space. Halmos [3] gave an example of a transitive lattice of subspaces with only 5 nontrivial elements, and in [4] (cf. [7, §4.7]) an example was constructed with only 4 nontrivial elements. It is easily seen that there is no transitive subspace lattice with only two nontrivial elements. The question remains: is there a transitive lattice of subspaces with only three nontrivial elements? We have not been able to answer this question, but we show (Theorem 2) that an affirmative answer would follow from the existence of a pair of operator ranges that are simultaneously left invariant by no nonscalar operator. In addition, we construct (Corollary 3) a pair of linear manifolds that are simultaneously left invariant by no nonscalar operator. In what follows, a collection of linear manifolds in a complex Banach space is transitive if the only operators (i.e., bounded linear transformations) leaving all of the manifolds of the collection invariant are the scalars. We use the term linear transformation to refer to a possibly unbounded transformation defined on the entire space. The word dimension always refers to algebraic (linear, Hamel) dimension. The symbol c denotes the cardinality of the continuum. Our methods are heuristically inspired by those of Shields [8]. Lemma. Suppose that X is a separable infinite-dimensional Banach space, M is a linear manifold in X, and A: X -» X is a linear transformation with the property that, for every x in X, the vector Ax is a multiple of x modulo M. Then there is a linear transformation F: X -M and a scalar X such that A -X + F. Also, if A is bounded and M has dimension less than c, then F has finite rank. Proof. If is clear from the hypothesis that A leaves M invariant. Since every nonzero vector is an eigenvector for the quotient operator on X/M induced by A,

doi:10.2307/2044366
fatcat:xocbvtk3gjdwhf5dj2xguikq7y