Spektriteoreem

Spektriteoreem, or the spectral theorem in its traditional form, is a foundational result in functional analysis that describes the structure of certain linear operators on Hilbert spaces. It provides a way to understand and manipulate operators by decomposing them into simpler, spectrum-based components.

In its standard formulation, the theorem applies to bounded normal operators on a complex Hilbert space. A normal operator T satisfies TT* = T*T. There exists a unique projection-valued measure E defined on the Borel sets of the spectrum of T such that T can be expressed as an integral T = ∫ λ dE(λ). This spectral decomposition leads to a functional calculus: for any bounded Borel function f, one defines f(T) = ∫ f(λ) dE(λ). Consequences include that self-adjoint operators have spectra contained in the real line, and unitary operators have spectra on the unit circle.

In finite dimensions, the theorem reduces to the familiar diagonalization: a normal matrix is unitarily diagonalizable, with its eigenvalues forming the spectrum. For compact normal operators, there is an orthonormal basis of eigenvectors, and the operator acts like a diagonal operator with eigenvalues tending to zero.

There is also a version for unbounded self-adjoint operators, where the spectral measure yields a decomposition on the real line and requires domain considerations. Applications span quantum mechanics, differential equations, signal processing, and operator theory, providing a versatile framework for spectral analysis and functional calculus.

See also: spectral measure, functional calculus, normal operator, self-adjoint operator, unitary operator.