Let $X$ be a Banach space.

a) What does it mean for a bounded linear map $T: X \rightarrow X$ to be compact?

b) Let $\mathcal{B}(X)$ be the Banach space of all bounded linear maps $S: X \rightarrow X$. Let $\mathcal{B}_{0}(X)$ be the subset of $\mathcal{B}(X)$ consisting of all compact operators. Show that $\mathcal{B}_{0}(X)$ is a closed subspace of $\mathcal{B}(X)$. Show that, if $S \in \mathcal{B}(X)$ and $T \in \mathcal{B}_{0}(X)$, then $S T, T S \in \mathcal{B}_{0}(X)$.

c) Let

$$X=\ell^{2}=\left\{x=\left(x_{1}, x_{2}, \ldots\right): x_{j} \in \mathbb{C} \quad \text { and } \quad\|x\|_{2}^{2}=\sum_{j=1}^{\infty}\left|x_{j}\right|^{2}<\infty\right\}$$

and $T: X \rightarrow X$ be defined by

$$(T x)_{k}=\frac{x_{k+1}}{k+1} .$$

Is $T$ compact? What is the spectrum of $T ?$ Explain your answers.