If $A$ is a normal operator on an infinite dimensional Hilbert space $H$, then $H$ is the direct sum of a countably infinite collection of subspaces that reduce $A$, all with the same infinite dimensional.
For this problem I thought about compact normal operator, In this case I can write $H$ as a countable directed sum of finite dimensional reducing subspace. Is it correct? while the question supposed to be a directed sum of infinite dimensional subspaces. So because of it I get confused.
I do not have any idea about it. Please help me about it. Thanks in advance.