Contractions with the bicommutant property

Takahashi, Katsutoshi

doi:10.1090/S0002-9939-1985-0766534-7

Contractions with the bicommutant property
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by Katsutoshi Takahashi

Proc. Amer. Math. Soc. 93 (1985), 91-95

DOI: https://doi.org/10.1090/S0002-9939-1985-0766534-7

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Abstract:

It is shown that if $T$ is a contraction for which there is an operator $W$ with dense range such that $WT = SW$ for some unilateral shift $S$, then $T$ has the bicommutant property, that is, the double commutant of $T$ is the weakly closed algebra generated by $T$ and the identity. As an example of such a contraction we have a contraction $T$ such that $I - {T^ * }T$ is of trace class and the spectrum of $T$ fills the unit disc.

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