Show that the operator is invertible

Solution 1:

The space of bounded linear operators on $X$, denoted $\mathcal{L}(X)$, is a Banach space, since $X$ is complete. The norm associated with $\mathcal{L}(X)$ is the operator norm, i.e. \begin{align*} \|T\| = \sup_{\|x\| = 1} \|Tx\|. \end{align*}

Recall that a normed vector space is complete if and only if every absolutely convergent series converges.

So consider the series \begin{align*} \sum_{n=0}^{\infty} \|T\|^n \end{align*} Since $\|T\| < 1$, this series converges, hence \begin{align*} \sum_{n=0}^{\infty} T^n \end{align*} converges in $\mathcal{L}(X)$ to some element, call it $S$. To show that $S = (I-T)^{-1}$, it suffices to show that $S(I-T) = I$. Observe, \begin{align*} S(I - T) &= \sum_{n=0}^{\infty} T^n (I - T) \\ &= \sum_{n=0}^{\infty} T^n - \sum_{n=0}^{\infty} T^{n+1} \\ &= \sum_{n=0}^{\infty} T^n - \sum_{n=1}^{\infty} T^{n} \\ &= T^0 \\ &= I \end{align*} Thus, $S = (I-T)^{-1}$.

Solution 2:

When the operator $a*I_d - b ad(Y)$ is invertible, $a$, $b$ - complex numbers, $I_d$ - identity operator, $ad(Y)$ - adjoint operator generated by matrix $Y$. All acts in some linear finite dimensional vector space. By "when" I mean necessary and sufficient conditions on $a$, $b$, $Y$.