07-28-2023-1223 - DECOMPOSITION OF SPECTRUM (FUNCTIONAL ANALYSIS) (DRAFT)

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

• a point spectrum, consisting of the eigenvalues of $T$;
• a continuous spectrum, consisting of the scalars that are not eigenvalues but make the range of $T-\lambda$ 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. 

https://en.wikipedia.org/wiki/Decomposition_of_spectrum_(functional_analysis)