Spectral Theorem - Bounded Self-adjoint Operators

Bounded Self-adjoint Operators

The next generalization we consider is that of bounded self-adjoint operators on a Hilbert space. Such operators may have no eigenvalues: for instance let A be the operator of multiplication by t on L2, that is

Theorem. Let A be a bounded self-adjoint operator on a Hilbert space H. Then there is a measure space (X, Σ, μ) and a real-valued essentially bounded measurable function f on X and a unitary operator U:H → L2_μ(X) such that

where T is the multiplication operator:

and

This is the beginning of the vast research area of functional analysis called operator theory. see also the spectral measure.

There is also an analogous spectral theorem for bounded normal operators on Hilbert spaces. The only difference in the conclusion is that now may be complex-valued.

An alternative formulation of the spectral theorem expresses the operator as an integral of the coordinate function over the operator's spectrum with respect to a projection-valued measure.

When the normal operator in question is compact, this version of the spectral theorem reduces to the finite-dimensional spectral theorem above, except that the operator is expressed as a linear combination of possibly infinitely many projections.