= Solution
For a complex <Banach space>, the <spectrum of a bounded operator> and <resolvent set> are
$$
\sigma(T)=\{\lambda\in\mathbb C:T-\lambda I\text{ has no bounded two-sided inverse}\},\qquad
\rho(T)=\mathbb C\setminus\sigma(T).
$$
The <bounded inverse theorem> says that a bounded bijection between <Banach spaces> automatically has a bounded inverse, so failure of bijectivity suffices here. The <point spectrum> is
$$
\sigma_p(T)=\{\lambda\in\mathbb C:\ker(T-\lambda I)\ne\{0\}\},
$$
the set of <eigenvalues>. For a real <Banach space>, these definitions use its complexification.
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