Discrete spectrum (mathematics)

In mathematics, specifically in spectral theory, a discrete spectrum of a closed linear operator is defined as the set of isolated points of its spectrum such that the rank of the corresponding Riesz projector is finite.

Definition

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A point   in the spectrum   of a closed linear operator   in the Banach space   with domain   is said to belong to discrete spectrum   of   if the following two conditions are satisfied:[1]

  1.   is an isolated point in  ;
  2. The rank of the corresponding Riesz projector   is finite.

Here   is the identity operator in the Banach space   and   is a smooth simple closed counterclockwise-oriented curve bounding an open region   such that   is the only point of the spectrum of   in the closure of  ; that is,  

Relation to normal eigenvalues

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The discrete spectrum   coincides with the set of normal eigenvalues of  :

 [2][3][4]

Relation to isolated eigenvalues of finite algebraic multiplicity

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In general, the rank of the Riesz projector can be larger than the dimension of the root lineal   of the corresponding eigenvalue, and in particular it is possible to have  ,  . So, there is the following inclusion:

 

In particular, for a quasinilpotent operator

 

one has  ,  ,  ,  .

Relation to the point spectrum

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The discrete spectrum   of an operator   is not to be confused with the point spectrum  , which is defined as the set of eigenvalues of  . While each point of the discrete spectrum belongs to the point spectrum,

 

the converse is not necessarily true: the point spectrum does not necessarily consist of isolated points of the spectrum, as one can see from the example of the left shift operator,   For this operator, the point spectrum is the unit disc of the complex plane, the spectrum is the closure of the unit disc, while the discrete spectrum is empty:

 

See also

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References

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  1. ^ Reed, M.; Simon, B. (1978). Methods of modern mathematical physics, vol. IV. Analysis of operators. Academic Press [Harcourt Brace Jovanovich Publishers], New York.
  2. ^ 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.
  3. ^ 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.