A bounded operator on Hilbert space(say $H$) $A$ is said to be reductive if every invariant subspace for $A$ reduces $A$. ( we say a subspace $V\subset H$ reduces $A$ if $A(V)\subset V$ and $A(V^\perp)\subset V^\perp$)
This definition is given in the book "A course in Functional Analysis" chapter IX section 9 (9.1 Definition).
In the book the next paragraph states that -every normal compact operator is reductive.
Somehow I was unable to find out an argument to prove this. I was able to find an argument for finite dimensional case. Some hint would also be very helpful.