## ::Decomposition of spectrum (functional analysis)

### ::concepts

''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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