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Normal eigenvalue

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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

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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 (for example, when it is finite-dimensional), it is called the generalized eigenspace of corresponding to the eigenvalue .

Definition of a normal eigenvalue

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An eigenvalue of a closed linear operator in the Banach space with domain is called normal (in the original terminology, corresponds to a normally splitting finite-dimensional root subspace), 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.[1][2][3]

Equivalent characterizations of normal eigenvalues

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Let be a closed linear densely defined operator in the Banach space . The following statements are equivalent[4](Theorem III.88):

  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.[1][2][3]

If is a normal eigenvalue, then the root lineal coincides with the range of the Riesz projector, .[3]

Relation to the discrete spectrum

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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]

Decomposition of the spectrum of nonselfadjoint operators

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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:

See also

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References

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  1. ^ a b Gohberg, I. C; Kreĭn, M. G. (1957). "Основные положения о дефектных числах, корневых числах и индексах линейных операторов" [Fundamental aspects of defect numbers, root numbers and indexes of linear operators]. Uspekhi Mat. Nauk [Amer. Math. Soc. Transl. (2)]. New Series. 12 (2(74)): 43–118.
  2. ^ a b Gohberg, I. C; Kreĭn, M. G. (1960). "Fundamental aspects of defect numbers, root numbers and indexes of linear operators". American Mathematical Society Translations. 13: 185–264. doi:10.1090/trans2/013/08.
  3. ^ a b c Gohberg, I. C; Kreĭn, M. G. (1969). Introduction to the theory of linear nonselfadjoint operators. American Mathematical Society, Providence, R.I.
  4. ^ Boussaid, N.; Comech, A. (2019). Nonlinear Dirac equation. Spectral stability of solitary waves. American Mathematical Society, Providence, R.I. ISBN 978-1-4704-4395-5.
  5. ^ Reed, M.; Simon, B. (1978). Methods of modern mathematical physics, vol. IV. Analysis of operators. Academic Press [Harcourt Brace Jovanovich Publishers], New York.