= Point spectrum
{title2=$\sigma_p(T)$}
For a <bounded linear operator> $T:X\to X$ on a complex <Banach space>, the point spectrum is precisely its set of <eigenvalues>:
$$
\sigma_p(T)=\{\lambda\in\mathbb C:\ker(T-\lambda I)\ne\{0\}\}.
$$
Here $I$ is the <identity operator>. Thus membership requires an actual nonzero <eigenvector>, whereas the <spectrum of a bounded operator> only requires failure of a <bounded inverse>. Every <eigenvalue> belongs to the <spectrum>, but the inclusion can be strict. The <multiplication operator> discussed below has an entire interval in its <spectrum> and empty <point spectrum>.https://www.math.hkust.edu.hk/~makyli/4063/fanotes2_150615.pdf#page=38[HKUST Math 4063 notes, chapter 4, section 3: spectrum and point spectrum]{ref}
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