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 , let be a weakly closed algebra of bounded operators on which contains the identity. is said to be *transitive* if no closed subspace of is invariant under . There are no known proper subalgebras of which are transitive. In this paper it is shown that the only transitive algebra which satisfies a certain condition is . Furthermore, a generalization of condition is given which characterizes those algebras with totally ordered lattice of invariant subspaces that are reflexive.

**[1]**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**, 10.1090/S0002-9904-1950-09346-X**[2]**William B. Arveson,*A density theorem for operator algebras*, Duke Math. J.**34**(1967), 635–647. MR**0221293****[3]**Bent Fuglede,*A commutativity theorem for normal operators*, Proc. Nat. Acad. Sci. U. S. A.**36**(1950), 35–40. MR**0032944****[4]**Heydar Radjavi and Peter Rosenthal,*On invariant subspaces and reflexive algebras*, Amer. J. Math.**91**(1969), 683–692. MR**0251569****[5]**D. Sarason,*Invariant subspaces and unstarred operator algebras*, Pacific J. Math.**17**(1966), 511–517. MR**0192365**

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DOI:
https://doi.org/10.1090/S0002-9939-1971-0283607-6

Keywords:
Algebra of operators,
reflexive algebra,
transitive algebra,
lattice of invariant subspaces

Article copyright:
© Copyright 1971
American Mathematical Society