Some counterexamples in nonselfadjoint algebras. (English) Zbl 0656.47036

Let T be a bounded operator on a separable Hilbert space H. The closure of polynomials of T and I in the weak operator topoloy is denoted by \({\mathcal W}(T)\); the closure of the same set in the weak* topology of \({\mathcal B}(H)\) is denoted by \({\mathcal A}(T)\). T is called reflexive if \({\mathcal W}(T)=Alg\;Lat\;T\). \(\{T\}'\) denotes the commutant of T.
The author shows by counterexamples that the equations \({\mathcal W}(T)=\{T\}'\cap Alg Lat T\) and \({\mathcal W}(T)={\mathcal A}(T)\) fail in general and that \(T\oplus T\) is not necessarily reflexive. It is shown that if \(T_ 1\) and \(T_ 2\) are reflexive, \(T_ 1\oplus T_ 2\) needs not to be reflexive, and that there is a reflexive T with \(T^*\) non-reflexive.
Reviewer: S.V.Kislyakov


47L30 Abstract operator algebras on Hilbert spaces
47A15 Invariant subspaces of linear operators
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