Difference between revisions of "Maximal spectral type"
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− | The type of the maximal spectral measure $\mu$ (i.e. its equivalence class) of a [[Normal operator|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|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. | + | The type of the maximal spectral measure $\mu$ (i.e. its equivalence class) of a [[Normal operator|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|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. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.I. Plesner, "Spectral theory of linear operators" , F. Ungar (1969) (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.I. Plesner, "Spectral theory of linear operators" , F. Ungar (1969) (Translated from Russian)</TD></TR></table> |
Latest revision as of 21:50, 1 January 2019
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.
References
[1] | A.I. Plesner, "Spectral theory of linear operators" , F. Ungar (1969) (Translated from Russian) |
Maximal spectral type. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Maximal_spectral_type&oldid=43650