MALAYSIAN MATHEMATICAL SOCIETY On Certain Representation of Topological Groups

In this note classes of groups representations of which have either invariant vectors or invariant functionals are introduced. Connection between these classes of groups is established. Let E be a separable topological vector space over the field of complex numbers C and GL(E) be the group of all linear automorphisms of E and G be a separable topological group and)(: EGLG →ρ be a linear representation of the group G in E. We denote by GE the subspace of invariant-)(Gρ elements in E, that is {}, allfor )(: GgxxgExE G ∈=∈ = ρ and denote by E ' the space of continuous linear functionals on the space E. A functional Ef ′ ∈ is called ρ(G)-invariant if f(x)x)f(ggf(x) 1 = = − for all.Gg ∈ A linear representation ρ of topological group G in E is called continuous if the mapping EEG → × defined by the formula xgxg)() ( ρ → is continuous. Everywhere we consider continuous representations. Definition 1. A linear representation ρ is called linear reductive (or tα-representation) if for each 0, ≠ ∈ xEx G there exists continuous ρ(G)-invariant linear functional Ef ′ ∈ such that.0) ( ≠xf In the case of}0{=GE the representation ρ is also called tα-representation. Definition 2. A linear representation ρ of the group G in E is said to be tβ-representation if for arbitrary nontrivial continuous ρ(G)-invariant functional Ef ′ ∈ there exists element GEx ∈ such that.0) ( ≠xf We recall that a closed subspace 1E in E is said to be t-complementable if there exists a closed subspace 2E in E such that}0{21 = ∩ EE and E is the topological direct sum of subspaces 1E and.2E Here one assumes that there exist continuous projections acting from E onto 1E and 2E [2]. In this case subspace 2E is called t-complementable to 1E and the notation 21 EEE ⊕ = is used