Invertibility of Identity Minus Operator
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Theorem
Suppose $B$ is a Banach space and $T \in \mathfrak{L}(B, B)$, the space of bounded linear operators on $B$. If $\| T \|_* < 1$ in the operator norm $\|\cdot\|_*$:, then $I - T$ is invertible and has inverse
$$(I - T)^{-1} = I + T + T^2 + T^3 + \ldots.$$
Proof
Define $S_n = I + T + T^2 + \ldots + T^n$. We first argue that $S_n$ converges to a bounded linear operator $S\in \mathfrak{L}(B,B)$. For any $n > m$,
| \(\ds \norm{S_n - S_m}_*\) | \(=\) | \(\ds \norm{T^{m+1} + \ldots + T^n}_*\) | ||||||||||||
| \(\ds \) | \(\leq\) | \(\ds \norm{T^{m+1} }_* + \ldots + \norm{T^n}_*\) | by Triangle Inequality and Operator Norm is Norm | |||||||||||
| \(\ds \) | \(\leq\) | \(\ds \norm{T}_*^{m+1} + \ldots + \norm{T}_*^n\) | by repeated use of Operator Norm on Banach Space is Submultiplicative on each term | |||||||||||
| \(\ds \) | \(\leq\) | \(\ds \sum_{j=m+1}^\infty \norm{T}_*^{j}\) | since each term is positive. |
Since $\|T\|_* < 1$ by assumption, this is a tail of a converging infinite series; therefore, by Tail of Convergent Series tends to Zero, this can be made arbitrarily small. Hence the sequence $S_n$ is in fact Cauchy; by Set of Bounded Linear Operators on Banach Space is Banach Space, it converges in the operator norm to some bounded linear $S = \lim_{n\to\infty} S_n$.
Next, expand $(I-T)S_k = (I-T)(I + T + \ldots + T^k)$ to yield $I - T^{k+1}$. We argue that $I - T^{k+1} \to I$ in norm:
| \(\ds \norm{ (I - T^{k+1}) - I }_*\) | \(=\) | \(\ds \norm{ - T^{k+1} }_*\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \norm { T^{k+1} }_*\) | by Operator Norm is Norm | |||||||||||
| \(\ds \) | \(\leq\) | \(\ds \norm { T }^{k+1}\) | by repeated application of Operator Norm on Banach Space is Submultiplicative | |||||||||||
| \(\ds \) | \(\to\) | \(\ds 0\) | since $\norm{T}_* < 1$ by assumption. |
Taking limits on the left-hand side, we get $\lim_{k\to\infty} (I-T)S_k = (I-T)S = I$.
A similar computation shows us that $\lim_{k\to\infty} S_k(I-T) = S(I-T) = I$. So $S=I+T+T^2+\ldots$ is the bounded two-sided inverse of $I-T$.
$\blacksquare$