# Is the adjoint of a quasinormal operator quasinormal as well?

I am trying to make sense of the various properties of operators on Hilbert spaces that generalise the notion of normality. It is known that for a (bounded) operator $A$ there are the following implications: $$A \text{ is normal } \Rightarrow A \text{ is quasinormal } \Rightarrow A \text{ is subnormal } \Rightarrow A \text{ is hyponormal }$$

If $A$ is normal, then clearly so is $A^*$. $A$ is hyponormal by definition if and only if $A^*A \geq AA^*$, so clearly if $A$ is hyponormal but not normal, then $A^*$ is not hyponormal.

The definition of quasinormal is that $A$ commutes with $A^*A$. If we instead require that $A$ commutes with $AA^*$, then we get (up to minor details) the definition of quasinormality of $A$. Thus, it would be quite convenient if $A^*$ was quasinormal whenever $A$ is quasinormal. Does that happen to be true?

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Consider the unilateral shift S. It clearly commutes with $S^*S = I$, but $S^*$ does not commute with the projection $SS^*$. It doesn't commute with any non-trivial projection, actually.