## On algebras of operators with totally ordered lattice of invariant subspaces

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- by John B. Conway PDF
- Proc. Amer. Math. Soc.
**28**(1971), 163-168 Request permission

## 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.

## References

- Emil Artin,
*The influence of J. H. M. Wedderburn on the development of modern algebra*, Bull. Amer. Math. Soc.**56**(1950), no. 1, 65–72. MR**1565174**, DOI 10.1090/S0002-9904-1950-09346-X - William B. Arveson,
*A density theorem for operator algebras*, Duke Math. J.**34**(1967), 635–647. MR**221293** - Bent Fuglede,
*A commutativity theorem for normal operators*, Proc. Nat. Acad. Sci. U.S.A.**36**(1950), 35–40. MR**32944**, DOI 10.1073/pnas.36.1.35 - Heydar Radjavi and Peter Rosenthal,
*On invariant subspaces and reflexive algebras*, Amer. J. Math.**91**(1969), 683–692. MR**251569**, DOI 10.2307/2373347 - D. Sarason,
*Invariant subspaces and unstarred operator algebras*, Pacific J. Math.**17**(1966), 511–517. MR**192365**

## Additional Information

- © Copyright 1971 American Mathematical Society
- 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