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'''Fact''' A bounded operator ''A'' is quasinormal if and only if in its [[polar decomposition]] ''A'' = ''UP'', the partial isometry ''U'' and positive operator ''P'' commute.
Given a quasinormal ''A'', the ideal is to construct dilations for ''U'' and ''P'' in a sufficiently nice way so everything commutes. Suppose for the moment that ''U'' is an isometry. Let ''V'' be the unitary dilation of ''U'',
:<math> V = \begin{bmatrix} U & I - UU^* \\ 0 & - U^* \end{bmatrix}
= \begin{bmatrix} U & D_{U^*} \\ 0 & - U^* \end{bmatrix}
.</math>
Define
:<math> Q = \begin{bmatrix} P & 0 \\ 0 & P \end{bmatrix}.</math>
The operator ''N'' = ''VQ'' is clearly an extension of ''A''; direct calculation shows that it is a normal extension.
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