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The structure of quasinormal operators and the double commutant property


Authors: John B. Conway and Pei Yuan Wu
Journal: Trans. Amer. Math. Soc. 270 (1982), 641-657
MSC: Primary 47B20; Secondary 47A65
DOI: https://doi.org/10.1090/S0002-9947-1982-0645335-6
MathSciNet review: 645335
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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 $ \{ \vert T{\vert^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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DOI: https://doi.org/10.1090/S0002-9947-1982-0645335-6
Article copyright: © Copyright 1982 American Mathematical Society

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