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 whereThe set of normal eigenvalues coincides with the discrete spectrum.of a linear operatorThis set is a linear manifold but not necessarily a vector space, since it is not necessarily closed inIf this set is closed (for example, when it is finite-dimensional), it is called the generalized eigenspace ofof a closed linear operatorin the Banach spaceis called normal (in the original terminology,corresponds to a normally splitting finite-dimensional root subspace), if the following two conditions are satisfied: That is, the restrictionbe a closed linear densely defined operator in the Banach spaceThe following statements are equivalent[4](Theorem III.88): Ifis a normal eigenvalue, then the root linealcoincides with the range of the Riesz projector,[3] 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.[5] The spectrum of a closed operatorin the Banach spacecan be decomposed into the union of two disjoint sets, the set of normal eigenvalues and the fifth type of the essential spectrum:

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