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 formula_1 has a bounded inverse.
The set of normal eigenvalues coincides with the discrete spectrum.
Root lineal.
Let formula_2 be a Banach space. The root lineal formula_3 of a linear operator formula_4 with domain formula_5 corresponding to the eigenvalue formula_6 is defined as
where formula_8 is the identity operator in formula_2.
This set is a linear manifold but not necessarily a vector space, since it is not necessarily closed in formula_2. If this set is closed (for example, when it is finite-dimensional), it is called the generalized eigenspace of formula_11 corresponding to the eigenvalue formula_12.
Definition.
An eigenvalue formula_6 of a closed linear operator formula_4 in the Banach space formula_2 with domain formula_16 is called "normal" (in the original terminology, "formula_12 corresponds to a normally splitting finite-dimensional root subspace"), if the following two conditions are satisfied:
That is, the restriction formula_28 of formula_11 onto formula_25 is an operator with domain formula_31 and with the range formula_32 which has a bounded inverse.
Equivalent definitions of normal eigenvalues.
Let formula_4 be a closed linear densely defined operator in the Banach space formula_2. The following statements are equivalent(Theorem III.88):
If formula_12 is a normal eigenvalue, then formula_53 coincides with the range of the Riesz projector, formula_54 (Gohberg–Krein 1969).
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 formula_4 in the Banach space formula_2 can be decomposed into the union of two disjoint sets, the set of normal eigenvalues and the fifth type of the essential spectrum: