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The name spectral theory was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic forms in infinitely many variables. The original spectral theorem was therefore conceived as a version of the theorem on principal axes of an ellipsoid , in an infinite-dimensional setting.
Hilbert spectral analysis is a signal analysis method applying the Hilbert transform to compute the instantaneous frequency of signals according to = (). After performing the Hilbert transform on each signal, we can express the data in the following form:
The study of spectra and related properties is known as spectral theory, which has numerous applications, most notably the mathematical formulation of quantum mechanics. The spectrum of an operator on a finite-dimensional vector space is precisely the set of eigenvalues. However an operator on an infinite-dimensional space may have additional ...
The spectral theorem has many variants. A particularly powerful version is as follows: Theorem. For any Abelian von Neumann algebra A on a separable Hilbert space H, there is a standard Borel space X and a measure μ on X such that it is unitarily equivalent as an operator algebra to L ∞ μ (X) acting on a direct integral of Hilbert spaces
In mathematics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the distribution and square-integrable aspects of functional analysis. Such spaces were introduced to study spectral theory.
In functional analysis, compact operators are linear operators on Banach spaces that map bounded sets to relatively compact sets. In the case of a Hilbert space H, the compact operators are the closure of the finite rank operators in the uniform operator topology. In general, operators on infinite-dimensional spaces feature properties that do ...
Throughout, is a fixed Hilbert space. A projection-valued measure on a measurable space (,), where is a σ-algebra of subsets of , is a mapping: such that for all , is a self-adjoint projection on (that is, () is a bounded linear operator (): that satisfies () = and () = ()) such that = (where is the identity operator of ) and for every ,, the function defined by (), is a complex measure on ...
In mathematics, particularly in operator theory, Wold decomposition or Wold–von Neumann decomposition, named after Herman Wold and John von Neumann, is a classification theorem for isometric linear operators on a given Hilbert space. It states that every isometry is a direct sum of copies of the unilateral shift and a unitary operator.