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.
The discrete spectrum can also be defined as the set of normal eigenvalues.

Definition

A point
in the spectrum of a closed linear operator in the Banach space with domain is said to belong to the 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 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,

Normal eigenvalues

The set of points in the discrete spectrum is equal to the set of normal eigenvalues.

Root lineal

Let be a Banach space. Consider a partially defined linear operator with domain. The root lineal corresponding to an eigenvalue is defined as the set of elements such that all belong to, and that after finitely many steps, we end up with zero:.
This set is a linear manifold but is not necessarily closed. If it is closed, it is called the generalized eigenspace of corresponding to the eigenvalue.

Normal eigenvalue

An eigenvalue of a closed linear operator in the Banach space with domain is called normal if the following two conditions are satisfied:
  1. The algebraic multiplicity of is finite:, where is the root lineal of corresponding to the eigenvalue ;
  2. The space can be decomposed into a direct sum, where is an invariant subspace of in which has a bounded inverse.

Relation to other spectra

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,.
Therefore, is an isolated eigenvalue of finite algebraic multiplicity, but it is not in the discrete spectrum:

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. Each point of the discrete spectrum is an eigenvalue, so
However, they may be unequal. An eigenvalue may not be an isolated point of the spectrum, or it may be isolated, but with an infinite-rank Riesz projector. For example, for the left shift operator,
the point spectrum is the open unit disc in the complex plane, the full spectrum is the closed unit disc, and the discrete spectrum is empty:
This is because has no isolated points.

Spectral decomposition

The spectrum of a closed operator in a Banach space can be decomposed into the union of two disjoint sets: the discrete spectrum and the fifth type of the essential spectrum :