Invertibility of Identity Minus Operator/Corollary

Corollary to Invertibility of Identity Minus Operator
Let $\GF \in \set {\R, \C}$.

Let $X$ be a Banach space over $\GF$. Let $T : X \to X$ be a invertible bounded linear operator.

Let $S : X \to X$ be a bounded linear operator such that:


 * $\norm S_{\map \BB X} \norm {T^{-1} }_{\map \BB X} < 1$

Then $T + S : X \to X$ is an invertible bounded linear operator.

Proof
We have:


 * $\norm {-S T^{-1} }_{\map \BB X} \le \norm S_{\map \BB X} \norm {T^{-1} }_{\map \BB X} < 1$

from, Norm on Bounded Linear Transformation is Submultiplicative.

From Invertibility of Identity Minus Operator, we have:


 * $I - \paren {-S T^{-1} } = I + S T^{-1}$ is an invertible bounded linear operator.

So from Composition of Bounded Linear Transformations is Bounded, we have that:


 * $\paren {I + S T^{-1} } T = T + S$ is a invertible bounded linear operator.