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A [[Hilbert space|Hilbert space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r0823401.png" /> containing a linear, everywhere-dense subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r0823402.png" />, on which the structure of a topological vector space is defined, such that the imbedding is continuous. This imbedding generates a continuous imbedding of the dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r0823403.png" /> and a chain of continuous imbeddings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r0823404.png" /> (using the standard identification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r0823405.png" />). The most interesting case is that in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r0823406.png" /> is a [[Nuclear space|nuclear space]]. The following strengthening of the spectral theorem for self-adjoint operators acting on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r0823407.png" /> is true: Any [[Self-adjoint operator|self-adjoint operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r0823408.png" /> mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r0823409.png" /> continuously (in the topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234010.png" />) onto itself possesses a complete system of generalized eigenfunctions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234011.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234012.png" /> is a set of indices), i.e. elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234013.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234014.png" />,
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A [[Hilbert space|Hilbert space]] $ \mathcal{H} $ containing a linear, everywhere-dense subset $ \Phi \subseteq \mathcal{H} $, on which the structure of a topological vector space is defined, such that the imbedding is continuous. This imbedding generates a continuous imbedding of the dual space $ \mathcal{H}' \subseteq \Phi' $ and a chain of continuous imbeddings $ \Phi \subseteq \mathcal{H} \subseteq \Phi' $ (using the standard identification $ \mathcal{H}' = \mathcal{H} $). The most interesting case is that in which $ \Phi $ is a [[Nuclear space|nuclear space]]. The following strengthening of the spectral theorem for self-adjoint operators acting on $ \mathcal{H} $ is true: Any [[Self-adjoint operator|self-adjoint operator]] $ A $ mapping $ \Phi $ continuously (in the topology of $ \Phi $) onto itself possesses a complete system of generalized eigenfunctions $ (F_{\alpha} \mid \alpha \in \mathfrak{A}) $ ($ \mathfrak{A} $ is a set of indices), i.e. elements $ F_{\alpha} \in \Phi' $ such that for any $ \phi \in \Phi $,
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234015.png" /></td> </tr></table>
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{F_{\alpha}}(A \phi) = \lambda_{\alpha} {F_{\alpha}}(\phi), \qquad \alpha \in \mathfrak{A},
 
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$$
where the set of values of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234017.png" />, is contained in the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234018.png" /> (cf. [[Spectrum of an operator|Spectrum of an operator]]) and has full measure with respect to the [[Spectral measure|spectral measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234021.png" />, of any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234022.png" />. The completeness of the system means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234023.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234025.png" />, for at least one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234026.png" />. Moreover, for any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234027.png" />, its expansion with respect to the system of generalized eigenfunctions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234028.png" /> exists and generalizes the known expansion with respect to the basis of eigenvectors for an operator with a discrete spectrum.
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where the set of values of the function $ \alpha \mapsto \lambda_{\alpha} $, $ \alpha \in \mathfrak{A} $, is contained in the spectrum of $ A $ (cf. [[Spectrum of an operator|Spectrum of an operator]]) and has full measure with respect to the [[Spectral measure|spectral measure]] $ {\sigma_{f}}(\lambda) $, $ f \in \mathcal{H} $, $ \lambda \in \Bbb{R} $, of any element $ f \in \mathcal{H} $. The completeness of the system means that $ {F_{\alpha}}(\phi) \neq 0 $ for any $ \phi \in \Phi $, $ \phi \neq 0 $, for at least one $ \alpha \in \mathfrak{A} $. Moreover, for any element $ \phi \in \Phi $, its expansion with respect to the system of generalized eigenfunctions $ (F_{\alpha} \mid \alpha \in \mathfrak{A}) $ exists and generalizes the known expansion with respect to the basis of eigenvectors for an operator with a discrete spectrum.
  
 
Example: The expansion into a [[Fourier integral|Fourier integral]]
 
Example: The expansion into a [[Fourier integral|Fourier integral]]
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234029.png" /></td> </tr></table>
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f(x) = \int_{\Bbb{R}} e^{i s x} \tilde{f}(s) ~ \mathrm{d}{s}, \qquad x \in \Bbb{R}, \qquad f,\tilde{f} \in {L^{2}}(\Bbb{R}),
 
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$$
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234030.png" /> is a system of generalized eigenfunctions of the differentiation operator, acting on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234031.png" />, arising under the natural rigging of this space by the Schwartz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234032.png" /> (cf. [[Generalized functions, space of|Generalized functions, space of]]). The same assertions are also correct for unitary operators acting on a rigged Hilbert space.
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$ (x \mapsto e^{i s x} \mid s \in \Bbb{R}) $ is a system of generalized eigenfunctions of the differentiation operator, acting on $ {L^{2}}(\Bbb{R}) $, arising under the natural rigging of this space by the Schwartz space $ \mathcal{S}(\Bbb{R}) $ (cf. [[Generalized functions, space of|Generalized functions, space of]]). The same assertions are also correct for unitary operators acting on a rigged Hilbert space.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.M. Gel'fand,   G.E. Shilov,   "Some problems in the theory of differential equations" , Moscow (1958) (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.M. Gel'fand,   N.Ya. Vilenkin,   "Generalized functions. Applications of harmonic analysis" , '''4''' , Acad. Press (1964) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> Yu.M. [Yu.M. Berezanskii] Berezanskiy,   "Expansion in eigenfunctions of selfadjoint operators" , Amer. Math. Soc. (1968) (Translated from Russian)</TD></TR></table>
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<table>
 
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<TR><TD valign="top">[1]</TD> <TD valign="top">I.M. Gel'fand, G.E. Shilov, “Some problems in the theory of differential equations”, Moscow (1958) (In Russian)</TD></TR>
 
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<TR><TD valign="top">[2]</TD> <TD valign="top">I.M. Gel'fand, N.Ya. Vilenkin, “Generalized functions. Applications of harmonic analysis”, '''4''', Acad. Press (1964) (Translated from Russian)</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">Yu.M. [Yu.M. Berezanskii] Berezanskiy, “Expansion in eigenfunctions of selfadjoint operators”, Amer. Math. Soc. (1968) (Translated from Russian)</TD></TR></table>
  
 
====Comments====
 
====Comments====
A rigged Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082340/r08234033.png" /> is also called a Gel'fand triple. Occasionally one also finds the phrases nested Hilbert space, or equipped Hilbert space.
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A rigged Hilbert space $ \Phi \subseteq \mathcal{H} \subseteq \Phi' $ is also called a Gel'fand triple. Occasionally one also finds the phrases nested Hilbert space, or equipped Hilbert space.

Latest revision as of 00:44, 16 September 2015

A Hilbert space $ \mathcal{H} $ containing a linear, everywhere-dense subset $ \Phi \subseteq \mathcal{H} $, on which the structure of a topological vector space is defined, such that the imbedding is continuous. This imbedding generates a continuous imbedding of the dual space $ \mathcal{H}' \subseteq \Phi' $ and a chain of continuous imbeddings $ \Phi \subseteq \mathcal{H} \subseteq \Phi' $ (using the standard identification $ \mathcal{H}' = \mathcal{H} $). The most interesting case is that in which $ \Phi $ is a nuclear space. The following strengthening of the spectral theorem for self-adjoint operators acting on $ \mathcal{H} $ is true: Any self-adjoint operator $ A $ mapping $ \Phi $ continuously (in the topology of $ \Phi $) onto itself possesses a complete system of generalized eigenfunctions $ (F_{\alpha} \mid \alpha \in \mathfrak{A}) $ ($ \mathfrak{A} $ is a set of indices), i.e. elements $ F_{\alpha} \in \Phi' $ such that for any $ \phi \in \Phi $, $$ {F_{\alpha}}(A \phi) = \lambda_{\alpha} {F_{\alpha}}(\phi), \qquad \alpha \in \mathfrak{A}, $$ where the set of values of the function $ \alpha \mapsto \lambda_{\alpha} $, $ \alpha \in \mathfrak{A} $, is contained in the spectrum of $ A $ (cf. Spectrum of an operator) and has full measure with respect to the spectral measure $ {\sigma_{f}}(\lambda) $, $ f \in \mathcal{H} $, $ \lambda \in \Bbb{R} $, of any element $ f \in \mathcal{H} $. The completeness of the system means that $ {F_{\alpha}}(\phi) \neq 0 $ for any $ \phi \in \Phi $, $ \phi \neq 0 $, for at least one $ \alpha \in \mathfrak{A} $. Moreover, for any element $ \phi \in \Phi $, its expansion with respect to the system of generalized eigenfunctions $ (F_{\alpha} \mid \alpha \in \mathfrak{A}) $ exists and generalizes the known expansion with respect to the basis of eigenvectors for an operator with a discrete spectrum.

Example: The expansion into a Fourier integral $$ f(x) = \int_{\Bbb{R}} e^{i s x} \tilde{f}(s) ~ \mathrm{d}{s}, \qquad x \in \Bbb{R}, \qquad f,\tilde{f} \in {L^{2}}(\Bbb{R}), $$ $ (x \mapsto e^{i s x} \mid s \in \Bbb{R}) $ is a system of generalized eigenfunctions of the differentiation operator, acting on $ {L^{2}}(\Bbb{R}) $, arising under the natural rigging of this space by the Schwartz space $ \mathcal{S}(\Bbb{R}) $ (cf. Generalized functions, space of). The same assertions are also correct for unitary operators acting on a rigged Hilbert space.

References

[1] I.M. Gel'fand, G.E. Shilov, “Some problems in the theory of differential equations”, Moscow (1958) (In Russian)
[2] I.M. Gel'fand, N.Ya. Vilenkin, “Generalized functions. Applications of harmonic analysis”, 4, Acad. Press (1964) (Translated from Russian)
[3] Yu.M. [Yu.M. Berezanskii] Berezanskiy, “Expansion in eigenfunctions of selfadjoint operators”, Amer. Math. Soc. (1968) (Translated from Russian)

Comments

A rigged Hilbert space $ \Phi \subseteq \mathcal{H} \subseteq \Phi' $ is also called a Gel'fand triple. Occasionally one also finds the phrases nested Hilbert space, or equipped Hilbert space.

How to Cite This Entry:
Rigged Hilbert space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rigged_Hilbert_space&oldid=13083
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article