The structure of quasinormal operators and the double commutant property

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 T satisfy the equation { T } = A ( T ) \{ T\} = \mathcal {A}(T) , the weakly closed algebra generated by T T and 1 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 T as multiplication by z z on a subspace of an L 2 {L^2} space whenever there is a vector f f such that { | T | k T j f : k , j ⩾ 0 } \{ |T{|^k}{T^j}f:\,k,\,j \geqslant 0\} has dense linear span. The second representation theorem applies to those pure quasinormal operators T T such that T ∗ T {T^{\ast }}T is invertible. The second of these representation theorems will be used to determine which quasinormal operators have the double commutant property.