Rigged Hilbert space

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.