How to find the inverse of a continuous linear operator

Question

Let $X$ be a Banach space and $T$ is in the set of bounded linear operators from $X$ to $X$ and norm of $T$ is less than 1. Use contraction mapping principle show $I-T$ is an isomorphism.(I can show it is one to one, but not onto and continuous inverse)

Answer 1

If $I -T$ is isomorphism if for all $y \in X$ there only one $x \in X$ such that $$x - Tx = y$$ or equivalently, $x$ is the only fixed point of map $f_y : X \to X$ given for $f_y(z) = Tz + y$. But, $$||f_y(z) - f_y(w)|| = ||Tz - Tw|| \leq ||T||||z - w||$$ that is, $f_y$ is a contraction and therefore has only one fixed point.