Identity Minus Compact Linear Operator on Banach Space has Index Zero

Theorem

Let $\Bbb F \in \set {\R, \C}$.

Let $\struct {X, \norm {\,\cdot\,}_X}$ be a Banach space over $\Bbb F$.

Let $I : X \to X$ be the identity mapping.

Let $C : X \to X$ be a compact linear operator.

Then $I - C$ is a Fredholm operator with the index:

$\map {\mathrm{ind} } {I - C} = 0$

Proof

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Sources

• 2002: Peter D. Lax: Functional Analysis: $21.1$: Basic Properties of Compact Maps