Abstract
We prove a variant of the well-known result that intertwiners for the bilateral shift on ℓ2(Z) are unitarily equivalent to multiplication operators on L2(T). This enables us to unify and extend fundamental aspects of rigidity theory for bar-joint frameworks with an abelian symmetry group. In particular, we formulate the symbol function for a wide class of frameworks and show how to construct generalised rigid unit modes in a variety of new contexts.
Highlights
Property has been utilised to obtain combinatorial characterisations of so-called forced and incidental rigidity for finite bar-joint frameworks in dimension 2. (See [12, 22].)
In [18], it is shown that the rigidity matrix for a periodic bar-joint framework gives rise to a Hilbert space operator which is unitarily equivalent to a multiplication operator MΦ
The set of points in Td where Φ has a non-zero kernel is known as the RUM spectrum and takes its name from the phenomenon of rigid unit modes (RUMs) in silicates and zeolites
Summary
Let Γ be a locally compact Hausdorff abelian group. Denote by L2(Γ) the Hilbert space of square integrable functions, i.e. Borel-measurable functions f : Γ → C such that,. Check that since T commutes with the multiplication operators of characteristic functions, it follows that supp(T f ) ⊆ supp(f ). Let γ ∈ Γ and let Mδγ be the multiplication operator on L2(Γ) by the scalar function δγ. L satisfies the commuting property DγL = LDγ for all γ ∈ Γ if and only if L is unitarily equivalent to a multiplication operator MΦ ∈ B(L2(Γ)) for some Φ ∈ L∞(Γ). Note that we identify the Hilbert spaces L2(Γ, X) and L2(Γ) ⊗ X; given any g ∈ L2(Γ), the function gk ∈ L2(Γ, X) defined by gk(γ) = g(γ)xk, is identified with the elementary tensor g ⊗ xk ∈ L2(Γ) ⊗ X.
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