Decomposition of spectrum (functional analysis)

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

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