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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On algebras of operators with totally ordered lattice of invariant subspaces


Author: John B. Conway
Journal: Proc. Amer. Math. Soc. 28 (1971), 163-168
MSC: Primary 47.35; Secondary 46.00
MathSciNet review: 0283607
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Abstract: For a Hilbert space $ \mathcal{H}$, let $ \mathcal{A}$ be a weakly closed algebra of bounded operators on $ \mathcal{H}$ which contains the identity. $ \mathcal{A}$ is said to be transitive if no closed subspace of $ \mathcal{H}$ is invariant under $ \mathcal{A}$. There are no known proper subalgebras of $ \mathcal{B}(\mathcal{H})$ which are transitive. In this paper it is shown that the only transitive algebra which satisfies a certain condition $ \beta $ is $ \mathcal{B}(\mathcal{H})$. Furthermore, a generalization of condition $ \beta $ is given which characterizes those algebras with totally ordered lattice of invariant subspaces that are reflexive.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0283607-6
PII: S 0002-9939(1971)0283607-6
Keywords: Algebra of operators, reflexive algebra, transitive algebra, lattice of invariant subspaces
Article copyright: © Copyright 1971 American Mathematical Society