nLab
spectrum of an operator

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

In the case when XX is a complex separable Hilbert space this theory is best established. Then the spectrum is the set of all λ\lambda in \mathbb{C} in which the resolvent? (AλI) 1(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 AA is a bounded linear operator on a complex separable Hilbert space, then the spectrum σ(A)\sigma(A) is a compact subset of \mathbb{C}. The set σ d(A)\sigma_d(A) of ordinary eigenvalues of AA is a subset of σ(A)\sigma(A) called the discrete spectrum of AA. In particular case when AA is a compact operator the spectrum consists of the discrete spectrum only, except for possible addition of point 00.

Created on November 23, 2010 18:33:54 by Zoran Škoda (161.53.130.104)