Show that $X= \ker((u-\lambda)^p) \oplus (u-\lambda)^p(X)$ if $u$ is compact.

Question

Consider the following theorem in Murphy's book "C*-algebras and operator theory"

Can someone explain why the marked lines are true? I think this must be a matter of pure linear algebra but I can't find out the specifics. I.e. my question, why is

$$X= \ker((u-\lambda)^p) + (u-\lambda)^p(X)$$Thanks!

Answer 1

This is from dimensional reasons. You have that the kernel and the image are linearly independent and that the complement of the image has dimension of the defect, which is the same dimension as the kernel.

More concretely, since $\ker(u-\lambda)^p$ and $(u-\lambda)^p(X)$ are disjoint you have that the map $$\ker(u-\lambda)^p\to X/(u-\lambda)^p(X)$$ is injective, but this is an injective map between vector spaces of the same dimension, hence a vectors space isomorphism. Thus $\ker(u-\lambda)^p$ is a complement of $(u-\lambda)^p(X)$.