Show that $I-A$ is invertible iff $a_{k,k} \neq 1$ for all $k$

Question

Let $p \in [1,\infty)$. Let $\sum_{k=0}^{\infty} \left( \sum_{l=0}^{\infty} |a_{k,l}|^q \right)^\frac pq < \infty$ for $qp=q+p$. And let $a_{k,l}=0$ for $l>k$. Define
$$A: \mathscr l^p \to \mathscr l^p; \quad A(x_k)_k=\left( \sum_{l=0}^\infty a_{k,l} x_l\right)_k.$$

Show that $I-A$ is invertible iff $a_{k,k} \neq 1$ for all $k$.

My idea was to show $\lVert A \rVert < 1$ and use the Neumann series.

Answer 1

Hint: Consider the sequences $e_k $ defined by $(e_k)_j = \delta_{jk}$. What is the action of $I - A$ on them?