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Operator theory Definition

Spectral theory

Builds on Operator theorySpectrum of an operatorEigenvalueEigenvector

Definition

Spectral theory is the part of operator theory that studies a linear operator TT on a Banach or Hilbert space through its spectrum σ(T)\sigma(T). This is the set of complex numbers λ\lambda for which T−λIT - \lambda I has no bounded inverse. It extends the theory of eigenvalues and eigenvectors of matrices. Its main topics are the location and the parts of the spectrum, the resolvent and functional calculus. It also contains the spectral theorems, which represent operators as multiplication operators.

Remarks

Example: for the diagonal operator D(xn)=(xn/n)D(x_n) = (x_n / n) on ℓ2\ell^2, spectral theory gives σ(D)={1/n:n≥1}∪{0}\sigma(D) = \set{1/n : n \ge 1} \cup \set{0}. Each 1/n1/n is an eigenvalue with eigenvector the nn-th unit vector, and 00 is in the spectrum without being an eigenvalue.

Non-example: the study of the solutions of a nonlinear equation such as u′′=u3u'' = u^3 is not spectral theory in this sense, since no linear operator and no spectrum are involved. Only its linearisation can be studied spectrally.

The word "spectrum" was used by David Hilbert before quantum mechanics. Only later did it turn out that the spectral lines of atoms correspond to differences of eigenvalues of a quantum Hamiltonian.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Spectral_theory
About this page
TagRQ39XYAG36
URIm2://RQ39XYAG36
Kinddefinition
WritingW0 stub
MSC47A10, 47B15, 47A60
DefinesSpectral theory
UsesOperator theory, Spectrum of an operator, Eigenvalue, Eigenvector
Changed2 November 2026

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