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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)" = ((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 L2 space whenever there is a vector f such that {I TIkTJf: k, j : 0) has dense linear span. The second representation theorem applies to those pure quasinormal operators T such that T*T is invertible. The second of these representation theorems will be used to determine which quasinormal operators have the double commutant property.

The Structure of Quasinormal Operators and the Double Commutant Property