In the case that $V$ is an infinite-dimensional complex separable Hilbert space, the (normal) eigenvalues only form the discrete spectrum. Instead, the full spectrum is the set of all $\lambda \in \mathbb{C}$ for which the resolvent$(A-\lambda I)^{-1}$ is not a bounded operator.

In other words, the spectrum is the complement of the subset of complex numbers for which the resolvent is a bounded operator.

The set $\sigma_d(A)$ of ordinary normaleigenvalues of $A$ is a subset of $\sigma(A)$ called the discrete spectrum of $A$. In particular case when $A$ is a compact operator the spectrum consists of the discrete spectrum only, except for possible addition of point $0$.