= Solution
For a bounded <linear operator> $T:H\to H$, the <spectrum> is
$$
\sigma(T)=\{\lambda\in\mathbb C:T-\lambda I\text{ has no bounded inverse}\},
$$
while the <point spectrum> is the set of <eigenvalue>[eigenvalues], equivalently the $\lambda$ for which $T-\lambda I$ is not injective.
A <compact operator> maps the unit ball to a relatively compact set. A <self-adjoint operator> satisfies $T=T^*$, or $\langle Tx,y\rangle=\langle x,Ty\rangle$ for all $x,y\in H$. A <finite-rank operator> has finite-dimensional image.
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