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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
DOI: https://doi.org/10.1090/S0002-9939-1971-0283607-6
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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Keywords: Algebra of operators, reflexive algebra, transitive algebra, lattice of invariant subspaces
Article copyright: © Copyright 1971 American Mathematical Society