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.