Example of a Bounded Linear Operator with Unbounded Spectrum.
For a very simple example, let $X$ be the polynomial ring $\mathbb{C}[x]$ and let $T$ be multiplication by $x$. Then $\lambda I-T$ is not invertible for any $\lambda\in\mathbb{C}$ (it is never surjective), so the spectrum of $T$ is all of $\mathbb{C}$.
All that remains is to find a norm on $X$ for which $T$ is bounded. This is easy: for instance, you could consider $\mathbb{C}[x]$ as a subspace of $C[0,1]$ (the restrictions of polynomial functions to $[0,1]$), and then clearly $\|T\|\leq 1$.