The structure of quasinormal operators and the double commutant property

Conway, John B.; Wu, Pei Yuan

doi:10.1090/S0002-9947-1982-0645335-6

The structure of quasinormal operators and the double commutant property
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Trans. Amer. Math. Soc. 270 (1982), 641-657 Request permission

Abstract:

In this paper a characterization of those quasinormal operators having the double commutant property is obtained. That is, a necessary and sufficient condition is given that a quasinormal operator $T$ satisfy the equation $\{ T\} '' = \mathcal {A}(T)$, the weakly closed algebra generated by $T$ and $1$. In particular, it is shown that every pure quasinormal operator has the double commutant property. In addition two new representation theorems for certain quasinormal operators are established. The first of these represents a pure quasinormal operator $T$ as multiplication by $z$ on a subspace of an ${L^2}$ space whenever there is a vector $f$ such that $\{ |T{|^k}{T^j}f: k, j \geqslant 0\}$ has dense linear span. The second representation theorem applies to those pure quasinormal operators $T$ such that ${T^{\ast }}T$ is invertible. The second of these representation theorems will be used to determine which quasinormal operators have the double commutant property.

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