Discrete spectrum (mathematics)


In mathematics, specifically in spectral theory, a discrete spectrum of a closed linear operator is defined as the set of isolated points of its spectrum such that the rank of the corresponding Riesz projector is finite.

Definition

A point
in the spectrum of a closed linear operator in the Banach space with domain is said to belong to discrete spectrum of if the following two conditions are satisfied:
  1. is an isolated point in ;
  2. The rank of the corresponding Riesz projector is finite.
Here is the identity operator in the Banach space and is a smooth simple closed counterclockwise-oriented curve bounding an open region such that is the only point of the spectrum of in the closure of ; that is,

Relation to normal eigenvalues

The discrete spectrum coincides with the set of normal eigenvalues of :

Relation to isolated eigenvalues of finite algebraic multiplicity

In general, the rank of the Riesz projector can be larger than the dimension of the root lineal of the corresponding eigenvalue, and in particular it is possible to have,. So, there is the following inclusion:
In particular, for a quasinilpotent operator
one has
,,

Relation to the point spectrum

The discrete spectrum of an operator is not to be confused with the point spectrum, which is defined as the set of eigenvalues of.
While each point of the discrete spectrum belongs to the point spectrum,
the converse is not necessarily true: the point spectrum does not necessarily consist of isolated points of the spectrum, as one can see from the example of the left shift operator,
For this operator, the point spectrum is the unit disc of the complex plane, the spectrum is the closure of the unit disc, while the discrete spectrum is empty: