For a linear operator on a finite-dimensional complex vector space , the spectrum of is simply the subset of the field of complex numbers consisting of eigenvalues of . The set of eigenvalues is however not the best invariant in the -dimensional case: it looks like generalized eigenvectors not belonging to (say in the sense of Gelfand triple) should be considered.
In the case when is a complex separable Hilbert space this theory is best established. Then the spectrum is the set of all in in which the resolvent? 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 is a bounded linear operator on a complex separable Hilbert space, then the spectrum is a compact subset of . The set of ordinary eigenvalues of is a subset of called the discrete spectrum of . In particular case when is a compact operator the spectrum consists of the discrete spectrum only, except for possible addition of point .