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===Quasinormal operators===
An operator ''A'' is said to be '''[[quasinormal operator|quasinormal]]''' if ''A'' commutes with ''A*A''.<ref name="Conway1991">{{citation|author=John B. Conway|title=The Theory of Subnormal Operators|url=https://books.google.com/books?id=Ho7yBwAAQBAJ|accessdate=15 June 2017|year=1991|publisher=American Mathematical Soc.|isbn=978-0-8218-1536-6|page=2|chapter=11}}</ref> A normal operator is thus quasinormal; the converse is not true. A counter example is given, as above, by the unilateral shift. Therefore, the family of normal operators is a proper subset of both quasinormal and subnormal operators. A natural question is how are the quasinormal and subnormal operators related.
We will show that a quasinormal operator is necessarily subnormal but not vice versa. Thus the normal operators is a proper subfamily of quasinormal operators, which in turn are contained by the subnormal operators. To argue the claim that a quasinormal operator is subnormal, recall the following property of quasinormal operators:
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