正規作用素のbicommutantについて

概要

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Let H be a complex Hilbert space. A bounded linear operator T on H is said to be reductive if every invariant subspace for T is reducing. The commutant of a set S of bounded linear operators on H is the collection of all bounded linear operators on H which commute with each operator in S. The commutant of the commutant of S is called the bicommutant of S. A bounded linear operator T on H is said to be of class (BC) if the bicommutant of T coincides with the weakly closed algebra generated by T and the identity operator of H. In this paper we prove that a bounded linear normal operator on H is of class (BC) if and only if it is reductive.
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