Let $X$ be a Banach space and suppose that $T: X \rightarrow X$ is a bounded linear operator. What is an eigenvalue of $T ?$ What is the spectrum $\sigma(T)$ of $T ?$

Let $X=C[0,1]$ be the space of continuous real-valued functions $f:[0,1] \rightarrow \mathbb{R}$ endowed with the sup norm. Define an operator $T: X \rightarrow X$ by

$$T f(x)=\int_{0}^{1} G(x, y) f(y) d y$$

where

$$G(x, y)= \begin{cases}y(x-1) & \text { if } y \leqslant x \\ x(y-1) & \text { if } x \leqslant y\end{cases}$$

Prove the following facts about $T$ :

(i) $T f(0)=T f(1)=0$ and the second derivative $(T f)^{\prime \prime}(x)$ is equal to $f(x)$ for $x \in(0,1)$;

(ii) $T$ is compact;

(iii) $T$ has infinitely many eigenvalues;

(iv) 0 is not an eigenvalue of $T$;

(v) $0 \in \sigma(T)$.

[The Arzelà-Ascoli theorem may be assumed without proof.]