Self-adjoint Operator - Spectral Multiplicity Theory

Spectral Multiplicity Theory

The multiplication representation of a self-adjoint operator, though extremely useful, is not a canonical representation. This suggests that it is not easy to extract from this representation a criterion to determine when self-adjoint operators A and B are unitarily equivalent. The finest grained representation which we now discuss involves spectral multiplicity. This circle of results is called the Hahn-Hellinger theory of spectral multiplicity.

We first define uniform multiplicity:

Definition. A self-adjoint operator A has uniform multiplicity n where n is such that 1 ≤ n ≤ ω if and only if A is unitarily equivalent to the operator M_f of multiplication by the function f(λ) = λ on

where H_n is a Hilbert space of dimension n. The domain of M_f consists of vector-valued functions ψ on R such that

Non-negative countably additive measures μ, ν are mutually singular if and only if they are supported on disjoint Borel sets.

Theorem. Let A be a self-adjoint operator on a separable Hilbert space H. Then there is an ω sequence of countably additive finite measures on R (some of which may be identically 0)

such that the measures are pairwise singular and A is unitarily equivalent to the operator of multiplication by the function f(λ) = λ on

This representation is unique in the following sense: For any two such representations of the same A, the corresponding measures are equivalent in the sense that they have the same sets of measure 0.

The spectral multiplicity theorem can be reformulated using the language of direct integrals of Hilbert spaces:

Theorem. Any self-adjoint operator on a separable Hilbert space is unitarily equivalent to multiplication by the function λ → λ on

The measure equivalence class of μ (or equivalently its sets of measure 0) is uniquely determined and the measurable family {H_x}_x is determined almost everywhere with respect to μ.