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Completely Reducible Operator Algebras and Spectral Synthesis

Published online by Cambridge University Press:  20 November 2018

Shlomo Rosenoer*
Affiliation:
Bol'shaya Serpuhovskaya ul., 31, korp. 6, app. 229A, Moscow, U.S.S.R.
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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 MH 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.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1982

References

1. T., Ando., A note on invariant subspaces of a compact normal operator, Arch. Math. 14 (1963), 337340./Google Scholar
2. Bochner, S. and J., von Neumann, Almost periodic functions in a group, II, Trans. Amer. Math. Soc. 37 (1935), 2150./Google Scholar
3. Dunford, N. and Schwartz, J. T., Linear operators, Part III: Spectral operators (Interscience, New York, 1971./Google Scholar
4. C.-K., Fong, Operator algebras with complemented invariant subspace lattices, Indiana Univ. Math. J. 26 (1977), 10451056./Google Scholar
5. Gillespie, T. A., Boolean algebras of projections and reflexive algebras of operators, Proc. London Math. Soc. 37 (1978), 5674./Google Scholar
6. Loginov, A. I. and V. S., Sul'man, On reductive operators and operator algebras, Izv. AN SSSR, Ser. Math 40 (1976), 845854 (Russian)./Google Scholar
7. Lomonosov, V. J., Invariant subspaces for the family of operators which commute with a completely continuous operator, Funct. Anal, and Appl. 7 (1973), 213214./Google Scholar
8. Nordgren, E. A., Radjavi, H. and Rosenthal, P., On operators with reducing invariant subspaces, Amer. J. Math. 97 (1975), 559570./Google Scholar
9. Rosenthal, P., On commutants of reductive operator algebras, Duke Math. J. 41 (1974), 829834./Google Scholar
10. Sarason, D. E., Invariant subspaces and unstarred operator algebras, Pacific J. Math. 17 (1966), 511517./Google Scholar