Search results for "Functional calculi"

Functional calculi for convolution operators on a discrete, periodic, solvable group

2009

Suppose T is a bounded self-adjoint operator on the Hilbert space L2(X,μ) and let T=∫SpL2TλdE(λ) be its spectral resolution. Let F be a Borel bounded function on [−a,a], SpL2T⊂[−a,a]. We say that F is a spectral Lp-multiplier for T, if F(T)=∫SpL2TF(λ)dE(λ) is a bounded operator on Lp(X,μ). The paper deals with l1-multipliers, where X=G is a discrete (countable) solvable group with ∀x∈G, x4=1, μ is the counting measure and TΦ:l2(G)∋ξ↦ξ∗Φ∈l2(G), where Φ=Φ∗ is a l1(G) function, suppΦ generates G. The main result of the paper states that there exists a Ψ on G such that all l1-multipliers for TΨ are real analytic at every interior point of Spl2(G)TΨ. We also exhibit self-adjoint Φ′s in l1(G) suc…