# why the restriction of a normal operator is normal?

can anyone explain why it's obvious that a normal operator restricted to a invariant subspace is still normal?

• Define the restricted operator $M = TP$ where $P$ is the projection into the invariant subspace. Can you show that $M^*M = MM^*$? – Cameron Williams Sep 18 at 10:38
• @cameron wlilliams sorry i cant. if the subspace is also T* invariant, i think the conclusion is obvious. but that's a corollary of what Artin prepare to prove. – ysTuan Sep 18 at 10:50
• Just use prop. 8.6.3 (b). If T is normal the equality holds for all vectors in your space, does it also hold for all vectors in your invariant subspace? – Dasi Sep 18 at 10:51
• @Dasi also has a good way to prove it. It's also the slightly easier way to do it. – Cameron Williams Sep 18 at 10:53
• @Dasi i cant even see why the subspace is invariant under the restriction of conjugate, so how it act on the subspace directly – ysTuan Sep 18 at 11:09

Let $$T_0:=T_{|W}$$. Now prove:
$$(T^*)_{|W}=T_0^*.$$