Normal eigenvalue


In mathematics, specifically in spectral theory, an eigenvalue of a closed linear operator is called normal if the space admits a decomposition into a direct sum of a finite-dimensional generalized eigenspace and an invariant subspace where has a bounded inverse.
The set of normal eigenvalues coincides with the discrete spectrum.

Root lineal

Let be a Banach space. The root lineal of a linear operator with domain corresponding to the eigenvalue is defined as
where is the identity operator in.
This set is a linear manifold but not necessarily a vector space, since it is not necessarily closed in. If this set is closed, it is called the generalized eigenspace of corresponding to the eigenvalue.

Definition

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 could be decomposed into a direct sum, where is an invariant subspace of in which has a bounded inverse.
That is, the restriction of onto is an operator with domain and with the range which has a bounded inverse.

Equivalent definitions of normal eigenvalues

Let be a closed linear densely defined operator in the Banach space. The following statements are equivalent:
  1. is a normal eigenvalue;
  2. is an isolated point in and is semi-Fredholm;
  3. is an isolated point in and is Fredholm;
  4. is an isolated point in and is Fredholm of index zero;
  5. is an isolated point in and the rank of the corresponding Riesz projector is finite;
  6. is an isolated point in, its algebraic multiplicity is finite, and the range of is closed..
If is a normal eigenvalue, then coincides with the range of the Riesz projector, .

Relation to the discrete spectrum

The above equivalence shows that the set of normal eigenvalues coincides with the discrete spectrum, defined as the set of isolated points of the spectrum with finite rank of the corresponding Riesz projector.

Decomposition of the spectrum of nonselfadjoint operators

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