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Completely Reducible Operator Algebras and Spectral Synthesis

Published online by Cambridge University Press:  20 November 2018

Shlomo Rosenoer
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 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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 5 , 01 October 1982 , pp. 1025 - 1035
Copyright
Copyright © Canadian Mathematical Society 1982

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