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- Gårding_domain abstract "In mathematics, a Gårding domain is a concept in the representation theory of topological groups. The concept is named after the mathematician Lars Gårding.Let G be a topological group and let U be a strongly continuous unitary representation of G in a separable Hilbert space H. Denote by g the family of all one-parameter subgroups of G. For each δ = { δ(t) | t ∈ R } ∈ g, let U(δ) denote the self-adjoint generator of the unitary one-parameter subgroup { U(δ(t)) | t ∈ R }. A Gårding domain for U is a linear subspace of H that is U(g)- and U(δ)-invariant for all g ∈ G and δ ∈ g and is also a domain of essential self-adjointness for UGårding showed in 1947 that, if G is a Lie group, then a Gårding domain for U consisting of infinitely differentiable vectors exists for each continuous unitary representation of G. In 1961, Kats extended this result to arbitrary locally compact topological groups. However, these results do not extend easily to the non-locally compact case because of the lack of a Haar measure on the group. In 1996, Danilenko proved the following result for groups G that can be written as the inductive limit of an increasing sequence G1 ⊆ G2 ⊆ ... of locally compact second countable subgroups:Let U be a strongly continuous unitary representation of G in a separable Hilbert space H. Then there exist a separable nuclear Montel space F and a continuous, bijective, linear map J : F → H such that the dual space of F, denoted by F∗, has the structure of a separable Fréchet space with respect to the strong topology on the dual pairing (F∗, F); the image of J, im(J), is dense in H; for all g ∈ G, U(g)(im(J)) = im(J); for all δ ∈ g, U(δ)(im(J)) ⊆ im(J) and im(J) is a domain of essential self-adjointness for U(δ); for all g ∈ G, J−1U(g)J is a continuous linear map from F to itself; moreover, the map G → Lin(F; F) taking g to J−1U(g)J is continuous with respect to the topology on G and the weak operator topology on Lin(F; F).The space F is known as a strong Gårding space for U and im(J) is called a strong Gårding domain for U. Under the above assumptions on G there is a natural Lie algebra structure on G, so it makes sense to call g the Lie algebra of G.".
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- Gårding_domain wikiPageRevisionID "423111225".
- Gårding_domain wikiPageWikiLink Bijection.
- Gårding_domain wikiPageWikiLink Category:Unitary_representation_theory.
- Gårding_domain wikiPageWikiLink Dense_set.
- Gårding_domain wikiPageWikiLink Direct_limit.
- Gårding_domain wikiPageWikiLink Dual_space.
- Gårding_domain wikiPageWikiLink Fréchet_space.
- Gårding_domain wikiPageWikiLink Generator_(mathematics).
- Gårding_domain wikiPageWikiLink Haar_measure.
- Gårding_domain wikiPageWikiLink Hilbert_space.
- Gårding_domain wikiPageWikiLink Inductive_limit.
- Gårding_domain wikiPageWikiLink Invariant_(mathematics).
- Gårding_domain wikiPageWikiLink Lars_Gårding.
- Gårding_domain wikiPageWikiLink Lie_algebra.
- Gårding_domain wikiPageWikiLink Lie_group.
- Gårding_domain wikiPageWikiLink Linear_map.
- Gårding_domain wikiPageWikiLink Linear_subspace.
- Gårding_domain wikiPageWikiLink Locally_compact_space.
- Gårding_domain wikiPageWikiLink Mathematician.
- Gårding_domain wikiPageWikiLink Mathematics.
- Gårding_domain wikiPageWikiLink Montel_space.
- Gårding_domain wikiPageWikiLink Nuclear_space.
- Gårding_domain wikiPageWikiLink One-parameter_group.
- Gårding_domain wikiPageWikiLink One-parameter_subgroup.
- Gårding_domain wikiPageWikiLink Representation_theory.
- Gårding_domain wikiPageWikiLink Second-countable_space.
- Gårding_domain wikiPageWikiLink Second_countable.
- Gårding_domain wikiPageWikiLink Self-adjoint.
- Gårding_domain wikiPageWikiLink Self-adjoint_operator.
- Gårding_domain wikiPageWikiLink Separable_space.
- Gårding_domain wikiPageWikiLink Strong_topology.
- Gårding_domain wikiPageWikiLink Strongly_continuous.
- Gårding_domain wikiPageWikiLink Subgroup.
- Gårding_domain wikiPageWikiLink Topological_group.
- Gårding_domain wikiPageWikiLink Unitary_representation.
- Gårding_domain wikiPageWikiLinkText "Gårding domain".
- Gårding_domain hasPhotoCollection Gårding_domain.
- Gårding_domain wikiPageUsesTemplate Template:Cite_journal.
- Gårding_domain wikiPageUsesTemplate Template:Orphan.
- Gårding_domain subject Category:Unitary_representation_theory.
- Gårding_domain comment "In mathematics, a Gårding domain is a concept in the representation theory of topological groups. The concept is named after the mathematician Lars Gårding.Let G be a topological group and let U be a strongly continuous unitary representation of G in a separable Hilbert space H. Denote by g the family of all one-parameter subgroups of G. For each δ = { δ(t) | t ∈ R } ∈ g, let U(δ) denote the self-adjoint generator of the unitary one-parameter subgroup { U(δ(t)) | t ∈ R }.".
- Gårding_domain label "Gårding domain".
- Gårding_domain sameAs m.02r7pd2.
- Gårding_domain sameAs Q5625932.
- Gårding_domain sameAs Q5625932.
- Gårding_domain wasDerivedFrom Gårding_domain?oldid=423111225.
- Gårding_domain isPrimaryTopicOf Gårding_domain.