nLab
spectrum of an operator

For a linear operator $A$ on a finite-dimensional complex vector space $X$, the spectrum of $A$ is simply the subset of the field of complex numbers consisting of eigenvalues of $A$. The set of eigenvalues is however not the best invariant in the $\mathrm{\infty }$-dimensional case: it looks like generalized eigenvectors not belonging to $X$ (say in the sense of Gelfand triple) should be considered.

In the case when $X$ is a complex separable Hilbert space this theory is best established. Then the spectrum is the set of all $\lambda$ in $ℂ$ in which the resolvent? $(A-\lambda I{)}^{-1}$ is not defined as 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.

If $A$ is a bounded linear operator on a complex separable Hilbert space, then the spectrum $\sigma (A)$ is a compact subset of $ℂ$. The set ${\sigma }_d(A)$ of ordinary eigenvalues 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 $0$.