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Journal of Operator Theory

Volume 44, Issue 1, Summer 2000  pp. 25-41.

The closure of the unitary orbit of the set of strongly irreducible operators in non-well ordered nest algebra

Authors You Qing Ji (1), Chun Lan Jiang (2), and Zong Yao Wang (3)
Author institution: (1) Department of Mathematics, Jilin University, Changchun, 130023, P.R. China
(2) Dept. of Applied Math. and Physics, Hebei University of Technology, Tianjin, 300103, P.R. China
(3) Department of Mathematics, East China University of Science and Technology, Shanghai, 200237, P.R. China

Summary:  A bounded linear operator $T$ on a Hilbert space $\cal H$ is strongly irreducible if $T$ does not commute with any non-trivial idempotent. A nest $\cal N$ is a chain of subspaces of $H$ contain $\{0\}$ and $\cal H$, which is closed under intersection and closed span. The nest algebra ${\rm alg}\, {\cal N}$ associated with $\cal N$ is the set of all operators which leave each subspace in $\cal N$ invariant. This paper proves that the norm closure of the unitary orbit of the strongly irreducible operators in a nest algebra is the set of operators whose spectrum is connected if and only if $\cal N$ or ${\cal N}^\perp$ are not well-ordered.

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