Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
up vote 3 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.