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.

Keywords: Strongly irreducible operator, nest, nest algebra, unitary orbit, spectrum

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