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 Axiom $\text N 2$: Positive Homogeneity, 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.

$\blacksquare$