Degenerate Linear Operator Plus Identity is Fredholm Operator

Theorem
Let $U$ be a vector space.

Let $T : U \to U$ be a degenerate linear operator.

Let $I_U : U \to U$ be the identity operator.

Then:
 * $T + I_U$

has finite index.

Proof
We need to show that both:
 * $\map \ker {T + I_U}$

and:
 * $U / {\Img {T + I_U} }$

are finite-dimensional.

Recall that:
 * $(1):\quad \map \dim {\Img T} < +\infty$

since $T$ is degenerate.

$\map \dim {\map \ker {T + I_U} } < +\infty$
Let $u \in \map \ker {T + I_U}$.

That is:
 * $Tu + u = 0$

Thus:
 * $u = \map T {-u} \in \Img T$

Hence:
 * $\map \ker {T + I_U} \subseteq \Img T$

Therefore:

$\map \dim {U / {\Img {T + I_U} } } < +\infty$
Observe that:
 * $\map \ker T \subseteq \Img {T + I_U}$

since if $u \in \map \ker T$, then:
 * $u = T u + u \in \Img {T + I_U}$

Thus: