Geodesic
Completely Reducible Operator Algebras and Spectral Synthesis
Canadian journal of mathematics, Tome 34 (1982) no. 5, pp. 1025-1035

Voir la notice de l'article provenant de la source Cambridge University Press

An algebra of bounded operators on a Hilbert space H is said to be reductive if it is unital, weakly closed and has the property that if M ⊂ H is a (closed) subspace invariant for every operator in , then so is M ⊥. Loginov and Šul'man [6] and Rosenthal [9] proved that if is an abelian reductive algebra which commutes with a compact operator K having a dense range, then is a von Neumann algebra. Note that in this case every invariant subspace of is spanned by one-dimensional invariant subspaces. Indeed, the operator KK * commutes with . Hence its eigenspaces are invariant for , so that H is an orthogonal sum of the finite-dimensional invariant subspaces of From this our claim easily follows. 
Rosenoer, Shlomo. Completely Reducible Operator Algebras and Spectral Synthesis. Canadian journal of mathematics, Tome 34 (1982) no. 5, pp. 1025-1035. doi: 10.4153/CJM-1982-074-9
@article{10_4153_CJM_1982_074_9,
     author = {Rosenoer, Shlomo},
     title = {Completely {Reducible} {Operator} {Algebras} and {Spectral} {Synthesis}},
     journal = {Canadian journal of mathematics},
     pages = {1025--1035},
     year = {1982},
     volume = {34},
     number = {5},
     doi = {10.4153/CJM-1982-074-9},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1982-074-9/}
}
TY  - JOUR
AU  - Rosenoer, Shlomo
TI  - Completely Reducible Operator Algebras and Spectral Synthesis
JO  - Canadian journal of mathematics
PY  - 1982
SP  - 1025
EP  - 1035
VL  - 34
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1982-074-9/
DO  - 10.4153/CJM-1982-074-9
ID  - 10_4153_CJM_1982_074_9
ER  - 
%0 Journal Article
%A Rosenoer, Shlomo
%T Completely Reducible Operator Algebras and Spectral Synthesis
%J Canadian journal of mathematics
%D 1982
%P 1025-1035
%V 34
%N 5
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1982-074-9/
%R 10.4153/CJM-1982-074-9
%F 10_4153_CJM_1982_074_9

[1] 1. T., Ando., A note on invariant subspaces of a compact normal operator, Arch. Math. 14 (1963), 337–340./ Google Scholar

[2] 2. Bochner, S. and J., von Neumann, Almost periodic functions in a group, II, Trans. Amer. Math. Soc. 37 (1935), 21–50./ Google Scholar

[3] 3. Dunford, N. and Schwartz, J. T., Linear operators, Part III: Spectral operators (Interscience, New York, 1971./ Google Scholar

[4] 4. C.-K., Fong, Operator algebras with complemented invariant subspace lattices, Indiana Univ. Math. J. 26 (1977), 1045–1056./ Google Scholar

[5] 5. Gillespie, T. A., Boolean algebras of projections and reflexive algebras of operators, Proc. London Math. Soc. 37 (1978), 56–74./ Google Scholar

[6] 6. Loginov, A. I. and V. S., Sul'man, On reductive operators and operator algebras, Izv. AN SSSR, Ser. Math 40 (1976), 845–854 (Russian)./ Google Scholar

[7] 7. Lomonosov, V. J., Invariant subspaces for the family of operators which commute with a completely continuous operator, Funct. Anal, and Appl. 7 (1973), 213–214./ Google Scholar

[8] 8. Nordgren, E. A., Radjavi, H. and Rosenthal, P., On operators with reducing invariant subspaces, Amer. J. Math. 97 (1975), 559–570./ Google Scholar

[9] 9. Rosenthal, P., On commutants of reductive operator algebras, Duke Math. J. 41 (1974), 829–834./ Google Scholar

[10] 10. Sarason, D. E., Invariant subspaces and unstarred operator algebras, Pacific J. Math. 17 (1966), 511–517./ Google Scholar

Cité par Sources :