On infinitely differentiable and Gevrey vectors for representations

On infinitely differentiable and Gevrey vectors for representations by A.F.M. ter Elst In the present paper we give a condition in order that the set of infinitely differentiable vectors for a representation 11 " in a Banach space is equal to the set of all infinitely differentiable vectors for the restriction of 11 " to a subgroup. Similar results for Gevrey vectors and analytic vectors are proved for unitary representations

Weighted strongly elliptic operators on Lie groups

AbstractLet (H, G, U) be a continuous representation of a Lie group G by bounded operators g ↦ U(g) on the Banach space X and let (X, g, dU) denote the representation of the Lie algebra g obtained by differentiation. If a1, …, ad′ is a Lie algebra basis of g, Ai = dU(ai) and Aα = Ai1 …Aik whenever α = (i1, …, ik) we consider the operators [formula] where the cα are complex coefficients satisfying a weighted strongly elliptic condition in which different directions may have different weights. This condition is such that the class of operators considered encompasses all the standard strongly elliptic operators. We prove that the closure H̄ of each such operator H generates a holomorphic semigroup S with holomorphy sector which contains a non-empty subsector determined by the coefficients and independent of the representation. Moreover, the semigroup S has a smooth representation independent kernel and we derive bounds on the kernel and all its derivatives. Finally we establish elliptic regularity properties for the operators and their powers and characterize the analytic and Gevrey vectors. As a corollary we derive optimal growth bounds for the eigenfunctions of the anharmonic oscillators P2m + Q2n

High order divergence-form elliptic operators on Lie groups

We give a straightforward proof that divergence-form elliptic operators of order m on a d-dimensional Lie group with m ≥ d have Hölder continuous kernels satisfying Gaussian bounds

Consistent operator semigroups and their interpolation

Under a mild regularity condition we prove that the generator of the interpolation of two C0-semigroups is the interpolation of the two generators