nLab
Gelfand triple

Contents

Idea

A Gel’fand triple is a structure that equips a Hilbert space with a dense topological vector subspace of good “test” functions, so that the dual of the subspace of test functions enhaces the Hilbert space by embedding it into a larger TVS whose elements can be considered as generalized eigenvectors for the continuous spectrum of normal (possibly unbounded) linear operators.

Definition

A Gel’fand triple is datum of the form

BJHKB * B\stackrel{J}\hookrightarrow H \stackrel{K}\hookrightarrow B^*

where HH is a separable Hilbert space, BB is a Banach space (or more general topological vector space (TVS)), B *B^* is a dual TVS of BB, J:BHJ:B\hookrightarrow H is an injective bounded operator with dense image, and KK is the composition of the canonical isomorphism HH *H\cong H^* determined by the inner product (i.e. given by Riesz theorem) and of the Banach transpose (dual) J *:H *B *J^*: H^*\to B^* of the operator JJ. The fact that J(B)J(B) is dense in HH implies that J *J^* (hence also KK) is injective as well.

Basic examples

A typical example is B=𝒮( n)B = \mathcal{S}(\mathbb{R}^n) (Schwarz space), H=L 2( n)H = L^2(\mathbb{R}^n) and B *=𝒮( n)B^* = \mathcal{S}'(\mathbb{R}^n) (the space of tempered (Schwarz) distributions). One of the basic facts on Fourier transform is that this Gel’fand triple is preserved by the Fourier transform.

Another natural example is 𝓁 1𝓁 2𝓁 \mathcal{l}^1\hookrightarrow \mathcal{l}^2\hookrightarrow \mathcal{l}^\infty.

A long example

Let HH be the Hilbert space L 2(,dx)L^2(\mathbb{R}, d x) consisting of square integrable functions ff with respect to Lebesgue measure. There is an unbounded self-adjoint operator

m x:HH:fxfm_x: H \to H: f \mapsto x \cdot f

where (xf)(y):=yf(y)(x \cdot f)(y) := y f(y). This operator is not defined on all of HH, but it is defined on a dense subspace of HH. For example, if SS is the Schwartz space consisting of smooth functions ff on \mathbb{R} all of whose derivatives f (n)(x)f^{(n)}(x) decay rapidly at infinity (more rapidly than any negative power of x|x|), then there is a dense inclusion map i:SHi: S \to H, and m xm_x is defined globally on SS.

Meanwhile the Schwartz space SS carries its own topology (as described in the article distribution), stronger than the topology it inherits from HH, and the space of tempered distributions S *S^* is defined to be the continuous dual of the TVS SS. Since the continuous inclusion i:SHi: S \to H is dense, it follows that any continuous functional

f:Sf: S \to \mathbb{C}

has at most one extension to a continuous functional HH \to \mathbb{C}. In other words, the adjoint map

i *:H *S *i^*: H^* \to S^*

is injective. In addition, the topology on SS is such that the operator m x:SSm_x: S \to S is continuous.

In this example, there is a dense inclusion SS *S \to S^* defined by the inner product pairing, and the operator m xm_x extends uniquely to an operator S *S *S^* \to S^*, called m xm_x by abuse of notation. Again, in this example, the operator m x:S *S *m_x: S^* \to S^* has an eigenvector s ξs_{\xi} for each ξ\xi \in \mathbb{R}:

m x(s ξ)=ξs ξm_x(s_{\xi}) = \xi s_{\xi}

(under construction)

Isomorphisms

An isomorphism of Gelfand triples (B 1,H 1,B 1 *)(B 2,H 2,B 2 *)(B_1,H_1,B^*_1)\to (B_2,H_2,B^*_2) is a unitary isomorphism H 1H 2H_1\to H_2 which restricts to an isomorphism of Banach spaces B 1B 2B_1\to B_2, and which extends to a weak**- and norm-preserving continuous isomorphism B 1 *B 2 *B_1^*\to B_2^*.

Terminology and references

Terminological variants and history

Usually, BB and B *B^* are Banach spaces, when we say Banach Gel’fand triple, there are some other variants involving more general topological vector spaces. In some cases one also uses the terminology rigged Hilbert space, following articles by Roberts and others since mid 1960-s. Nuclear Gel’fand triples are very common and then the notion is well behaved. In Russian literature the term enriched Hilbert space is used (оснащенное гильбертово пространство), sometimes translated also as equipped Hilbert space. The enriched word here is the same as in the phrase enriched category.

Early references

Gel’fand triples were introduced by Gel’fand school about 1955 and quickly incorporated into the theory of generalized functions.

  • [[I. M. Gelʹfand, A. G. Kostyučenko, Expansion in eigenfunctions of differential and other operators) (Russian), Dokl. Akad. Nauk SSSR (N.S.) 103 (1955), 349–352, MR73136

  • I. M. Gel'fand, N. Ja. Vilenkin, Generalized functions, vol. 4. Some applications of harmonic analysis. Equipped Hilbert spaces, Fizmatgiz, Moscow, 1961 MR146653, English transl. Acad. Press 1964 MR173945

Related early works include

  • Ciprian Foiaş, Décompositions intégrales des familles spectrales et semi-spectrales en opérateurs qui sortent de l’espace hilbertien, Acta Sci. Math. Szeged 20, 1959, 117–-155. MR115092

John Roberts continued the study in the context of quantum mechanics (in his thesis work suggested by Paul Dirac), changing the name to rigged.

  • John Roberts, The Dirac bra and ket formalism, J. Mathematical Phys. 7 (1966), 1097–1104, MR216836, Rigged Hilbert spaces in quantum mechanics , Comm. Math. Physics 3, n. 2 (1966), 98–119, MR228243, euclid

A unification of various inequivalent approaches is claimed in

  • M. Gadella, F. Gómez, A unified mathematical formalism for the Dirac formulation of quantum mechanics Foundations of Physics 32, No. 6, (2002)

  • S. Wickramasekara, A. Bohm, Symmetry representations in the rigged Hilbert space formulation of quantum mechanics, math-ph/0302018

Revised on December 1, 2013 12:51:21 by John Dougherty? (108.72.3.0)