General Self-adjoint Operators
Many important linear operators which occur in analysis, such as differential operators, are unbounded. There is also a spectral theorem for self-adjoint operators that applies in these cases. To give an example, any constant coefficient differential operator is unitarily equivalent to a multiplication operator. Indeed the unitary operator that implements this equivalence is the Fourier transform; the multiplication operator is a type of Fourier multiplier.
In general, spectral theorem for self-adjoint operators may take several equivalent forms.
Spectral theorem in the form of multiplication operator. For each self-adjoint operator T acting in a Hilbert space H, there exists a unitary operator, making an isometrically isomorphic mapping of the Hilbert space H onto the space L2(M, μ), where the operator T is represented as a multiplication operator.
The Hilbert space H where a self-adjoint operator T acts may be decomposed into a direct sum of Hilbert spaces Hi, in such a way that the operator T, narrowed to each space Hi, has a simple spectrum. It is possible to construct unique such decomposition (up to unitary equivalence), which is called an ordered spectral representation.
Read more about this topic: Spectral Theorem
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