Geodesic
On reductive operators and operator algebras
Izvestiya. Mathematics , Tome 10 (1976) no. 4, pp. 799-807.

Voir la notice de l'article provenant de la source Math-Net.Ru

We prove a theorem on the structure of weakly closed reductive operator algebras. The proof essentially relies on a known result of V. I. Lomonosov on transitive operator algebras containing a nonzero compact operator. We deduce a number of corollaries which apply to the reductivity problem. Bibliography: 20 titles. 
@article{IM2_1976_10_4_a7,
     author = {A. I. Loginov and V. S. Shulman},
     title = {On reductive operators and operator algebras},
     journal = {Izvestiya. Mathematics },
     pages = {799--807},
     publisher = {mathdoc},
     volume = {10},
     number = {4},
     year = {1976},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1976_10_4_a7/}
}
TY  - JOUR
AU  - A. I. Loginov
AU  - V. S. Shulman
TI  - On reductive operators and operator algebras
JO  - Izvestiya. Mathematics 
PY  - 1976
SP  - 799
EP  - 807
VL  - 10
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1976_10_4_a7/
LA  - en
ID  - IM2_1976_10_4_a7
ER  - 
%0 Journal Article
%A A. I. Loginov
%A V. S. Shulman
%T On reductive operators and operator algebras
%J Izvestiya. Mathematics 
%D 1976
%P 799-807
%V 10
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1976_10_4_a7/
%G en
%F IM2_1976_10_4_a7
A. I. Loginov; V. S. Shulman. On reductive operators and operator algebras. Izvestiya. Mathematics , Tome 10 (1976) no. 4, pp. 799-807. http://geodesic.mathdoc.fr/item/IM2_1976_10_4_a7/

[1] Wermer J., “On invariant subspaces of normal operators”, Proc. Amer. Math. Soc., 2 (1952), 270–277 | DOI | MR

[2] Dyer J., Pedersen E., Porcelli P., “An equivalent formulation of the invariant subspace conjecture”, Bull. Amer. Math. Soc., 78:6 (1972), 1020–1023 | DOI | MR | Zbl

[3] Lomonosov V. I., “Ob invariantnykh podprostranstvakh semeistva operatorov, kommutiruyuschikh s vpolne nepreryvnym”, Funkts. anal. i ego pril., 7:3 (1973), 55–56 | MR | Zbl

[4] Ando T., “A note on invariant subspaces of a compact normal operator”, Arch. Math., 14:4–5 (1963), 337–340 | DOI | MR | Zbl

[5] Rosenthal P., “Completely reducible operators”, Proc. Amer. Math. Soc., 19:4 (1968), 826–830 | DOI | MR | Zbl

[6] Nordgren E., Rosenthal P., “Algebras, containing unilateral shifts or finite-rank operators”, Duke Math. J., 40 (1973), 419–424 | DOI | MR | Zbl

[7] Nordgren E., Radjavi H., Rosenthal P., “Compact perturbations of normal operators with reducing invariant subspaces”, Notices Amer. Math. Soc., 20 (1973), A155

[8] Radjavi H., Rosenthal P., “On reflexive algebras of operators”, Indiana Univ. Math. J., 20:10 (1971), 935–937 | DOI | MR | Zbl

[9] Radjavi H., Rosenthal P., “A sufficient condition that an operator algebra be selfadjoint”, Can. J. Math., 23:4 (1971), 558–597 | MR

[10] Moore L., “Properties of reductive operators”, Notices Amer. Math. Soc., 20 (1973), A159

[11] Hoover T., “Operator algebras with reducing invariant subspaces”, Pacif. J. Math., 44:1 (1973), 173–179 | MR | Zbl

[12] Hoover T., “Operator algebras with complemented invariant subspace lattices”, Indiana Univ. Math. J., 22:11 (1973), 1029–1035 | DOI | MR | Zbl

[13] Fuglede B., “A commutativity theorem for normal operators”, Proc. Amer. Math. Soc., 36 (1950), 35–40 | MR | Zbl

[14] Barnes B., “Irreducible algebras of operators wich contain a minimal idempotent”, Proc. Amer. Math. Soc., 30:2 (1971), 337–343 | DOI | MR

[15] Saroson D., “Invariant subspaces and unstarred operator algebras”, Pacif. J. Math., 17 (1966), 511–517 | MR

[16] Shulman V. S., “O refleksivnykh operatornykh algebrakh”, Matem. sb., 87:2 (1972), 179–187

[17] Shulman V. S., “Teorema Faglida i refleksivnost”, Dokl. AN SSSR, 210:3 (1973), 543–544

[18] Loginov A. I., Shulman V. S., “O reduktivnykh operatornykh algebrakh i probleme invariantnogo podprostranstva”, Dokl. AN SSSR, 216:1 (1974), 36–38 | MR | Zbl

[19] Suzuki N., “Reduction theory of operators on Hilbert space”, Indiana Univ. Math. J., 20:10 (1971), 953–958 | DOI | MR | Zbl

[20] Sekefalvi-Nad B., Foyash Ch., Garmonichnyi analiz operatorov v gilbertovom prostranstve, Mir, M., 1970