Category:Invertibility of Identity Minus Operator

This category contains pages concerning 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 bounded linear operator such that:

$\norm T_{\map \BB X} < 1$

where $\norm {\, \cdot \,}_{\map \BB X}$ denotes the norm of a bounded linear operator.

Then $I - T$ is invertible as a bounded linear operator.

In particular:

$\ds \paren {I - T}^{-1} = \sum_{n \mathop = 0}^\infty T^n$