Overview
For the purposes of spectral theory, an operator is defined as a linear function from a vector space to itself.
{% T: E \rightarrow E %}
Definition of Spectrum
For a linear operator {% T %}, the spectrum of {% T %} (denoted {% \sigma(T) %}) is defined to be the set of all complex numbers {% \lambda %} for which {% T - \lambda I %} is not invertible. (sometimes stated that {% T - \lambda I %} is not an invertible bounded operator)
Spectrum and Eigenvalues
When the operator has an eigenvalue
{% Tx = \lambda x %}
then {% T - \lambda I %} is not invertible. That is, all eigenvalues are in the spectrum of the operator.
When the domain is finite dimensional, the spectrum consists entirely of the eigenvalues.
(see spectral decomposition
of a matrix)
In the case of an infinite dimensional space, there may not be any eigenvalues. However, there may exist a {% \lambda %} (or multiplie) and a sequence {% \psi_n %} such that
{% (T - \lambda)\psi_n \rightarrow 0 %}
These {% \lambda %} 's form a set of values in the spectrum, often referred to as the
"continuous" spectrum.