= Solution
Writing $\mathcal B(X)$ for the <Banach space of bounded linear operators> on $X$, the <spectrum> is
$$
\boxed{\sigma(T)=\{\lambda\in\mathbb C:\lambda I-T\text{ has no bounded two-sided inverse in }\mathcal B(X)\}.}
$$
Here $I$ is the <identity operator>. Since $X$ is a <Banach space>, the <bounded inverse theorem> makes this equivalent to failure of bijectivity of $\lambda I-T$.
The <spectral radius> is
$$
\boxed{r(T)=\sup\{|\lambda|:\lambda\in\sigma(T)\}.}
$$
For a nonzero complex <Banach space>, the <spectrum> of a <bounded linear operator> is nonempty and <compact>, so the <supremum> is a maximum. If the zero space is allowed, its empty <spectrum> is assigned <spectral radius> zero.
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