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::Decomposition of spectrum (functional analysis)

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''T''::spectrum    Operator::lambda    Minus::''h''    Bounded::space    Sigma::''x''    Measure::''h''

The spectrum of a linear operator <math>T</math> that operates on a Banach space <math>X</math> (a fundamental concept of functional analysis) consists of all scalars <math>\lambda</math> such that the operator <math>T-\lambda</math> does not have a bounded inverse on <math>X</math>. The spectrum has a standard decomposition into three parts:

  • a point spectrum, consisting of the eigenvalues of <math>T</math>
  • a continuous spectrum, consisting of the scalars that are not eigenvalues but make the range of <math>T-\lambda</math> a proper dense subset of the space;
  • a residual spectrum, consisting of all other scalars in the spectrum

This decomposition is relevant to the study of differential equations, and has applications to many branches of science and engineering. A well-known example from quantum mechanics is the explanation for the discrete spectral lines and the continuous band in the light emitted by excited atoms of hydrogen.


Decomposition of spectrum (functional analysis) sections
Intro   Definitions    Self adjoint operators on Hilbert space    See also    References   

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