Spectral Theorem - General Self-adjoint Operators

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 H_i, in such a way that the operator T, narrowed to each space H_i, has a simple spectrum. It is possible to construct unique such decomposition (up to unitary equivalence), which is called an ordered spectral representation.