It is known that every operator on a (separable) Hilbert space is the direct integral of irreducible operators, but not every one is the direct sum of irreducible ones. We show that an operator can have either finitely or uncountably many reducing subspaces, and the former holds if and only if the operator is the direct sum of finitely many irreducible operators no two of which are unitarily equivalent. We also characterize operators T which are direct sums of irreducible operators in terms of the C*-structure of the commutant of the von Neumann algebra generated by T.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm155-1-3,
author = {Jun Shen Fang and Chun-Lan Jiang and Pei Yuan Wu},
title = {Direct sums of irreducible operators},
journal = {Studia Mathematica},
volume = {157},
year = {2003},
pages = {37-49},
zbl = {1033.47006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm155-1-3}
}
Jun Shen Fang; Chun-Lan Jiang; Pei Yuan Wu. Direct sums of irreducible operators. Studia Mathematica, Tome 157 (2003) pp. 37-49. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm155-1-3/