Subnormal operator: Difference between revisions

Revision as of 08:30, 14 November 2006

Revision as of 20:27, 14 November 2006

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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 idealidea 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}