Maximal spectral type

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The type of the maximal spectral measure $\mu$ (i.e. its equivalence class) of a normal operator $A$ acting on a Hilbert space $H$. This measure is defined (up to equivalence) by the following condition. Let $E(\lambda)$ be the resolution of the identity in the spectral representation of the normal operator $A=\int\lambda\,dE(\lambda)$, and let $E(\Lambda)=\int_\Lambda dE(\lambda)$ (where $\Lambda$ denotes a Borel set) be the associated "operator-valued" measure. Then $E(\Lambda)=0$ precisely for those $\Lambda$ for which $\mu(\Lambda)=0$. Any $x\in H$ has an associated spectral measure $\mu_x(\Lambda)=(x,E(\Lambda)x)$; in these terms the definition of $\mu$ implies that for any $x$ the measure $\mu_x$ is absolutely continuous with respect to $\mu$ and there is an $x_0$ for which $\mu_{x_0}$ is equivalent to $\mu$ (that is, $x_0$ has maximal spectral type). If $H$ is separable, then a measure $\mu$ with these properties always exists, but if $H$ is not separable, then there is no such measure and $A$ does not have maximal spectral type. This complicates the theory of unitary invariants of normal operators in the non-separable case.


[1] A.I. Plesner, "Spectral theory of linear operators" , F. Ungar (1969) (Translated from Russian)
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Maximal spectral type. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article