Spectral type

Hellinger type, of a measure

The equivalence class with respect to mutual absolute continuity in the set of all non-negative measures on a given $\sigma$-algebra containing this measure. The set of spectral types with the order relation induced by the relation of absolute continuity of measures is a complete distributive lattice in which every countable subset is bounded.

The theory of spectral types is used in constructing a system of unitary invariants of normal operators. Let $A$ be an arbitrary normal operator on a space $H$ and let $E_A({\cdot})$ be the corresponding spectral measure in the plane. The spectral type $\sigma_A(x)$ of a vector $x \in H$ is defined as the spectral type of the measure $(E_A({\cdot})x,\, x)$. All spectral types of the form $\sigma_A(x)$ are called subordinate to the operator $A$. If $H$ is separable, then among the spectral types subordinate to $A$ there is a maximal one. In particular, all cyclic normal operators have a maximal spectral type. It turns out that a cyclic normal operator is determined by its maximal spectral type up to unitary equivalence. In the general case the system of unitary invariants includes the multiplicities of the homogeneous components of the maximal spectral type.

References

[1]	A.I. Plesner, "Spectral theory of linear operators" , F. Ungar (1965) (Translated from Russian)
[2]	J. Dixmier, "Les algèbres d'opérateurs dans l'espace hilbertien: algèbres de von Neumann" , Gauthier-Villars (1957)