nLab
spectrum of an operator

Contents

Contents

Idea

For a linear operator AA on a finite-dimensional complex vector space VV, the spectrum of AA is simply the subset of the complex numbers consisting of the eigenvalues of AA.

In the case that VV 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 λ\lambda \in \mathbb{C} for which the resolvent (AλI) 1(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.

Properties

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 normal 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.

References

See also

[[!redirects discrete spectra]

Last revised on May 28, 2021 at 06:34:44. See the history of this page for a list of all contributions to it.