Analytic and quasi-analytic vectors

Guided by researches in function theory, operator theorists gave the analogue to quasi-analytic classes. Let A be an operator in a Banach space X. A is not necessarily bounded hence the domain D(A) is not necessarily to be the whole space. We say x∈X is a C∞ vector if x∈⋂n≥1D(An). This is quite intuitive if we consider the differential operator. A vector is analytic if the series ∞∑n=0∥Anx∥tnn! has a positive radius of convergence. Finally, we say x is quasi-analytic for A provided that ∞∑n=0(1∥Anx∥)1/n=∞ or equivalently its nondecreasing majorant. Interestingly, if A is symmetric, then ∥Anx∥ is log convex.

Based on the density of quasi-analytic vectors, we have an interesting result.

(Theorem) Let A be a symmetric operator in a Hilbert space H. If the set of quasi-analytic vectors spans a dense subset, then A is essentially self-adjoint.

This theorem can be considered as a corollary to the fundamental theorem of quasi-analytic classes, by applying suitable Banach space techniques in lieu.

Hamburger moment problem

For a positive sequence {an}, we see it is the moment of a positive measure μ, i.e. an=∫Rtndμ(t) if and only if it is positively definite (proof). But the uniqueness is not guaranteed. Here we have a sufficient condition for this - using the concept of quasi-analytic vector. This is a old theorem (1922) but we are using operator theory to prove it which appeared decades later.

(Carleman's condition) Suppose {an} is the moment sequence of a positive measure μ on R, then μ is uniquely determined provided that ∑a−1/2n2n=∞.

Proof. Consider the Hilbert space H=L2(R,γ) and the operator A:f(t)↦tf(t). It is clear that A is self-adjoint. We shall work on the constant function u(t)≡1∈H. Since Anu=tn, we see u∈C∞, otherwise an is not defined. On the other hand, we have (Anu,u)=an⟹(A2nu,u)=∥Anu∥2=|(Anu,u)|2=a2n. But a−1/2n2n=∥Anu∥−1/n and as a result we see ∑a−1/2n2n=∑∥Anu∥−1/n=∞, hence u is quasi-analytic. In general, tn=Anu is quasi-analytic for all n≥0. Consider the space of polynomial P[t] with closure H1. It follows from the theorem above that A1=A|P[t] is essentially self-adjoint in H1. Hence H1 is invariant under the one-parameter group eiAs. Pick y∈P[t]⊥, we see (y,eiAsu)=∫Re−isty(t)dγ(t)=0, which implies that y=0 a.e. [γ]. It follows that H1=H or equivalently P[t] is dense in H. Suppose now we have another generating measure ν of {an}. With respect to ν, P[t] is still a dense space. But the norm on P[t] is fixed by {an}, hence we obtain an isometry between P[t]γ and P[t]ν, which extends to the isometry between L2(R,γ) and L2(R,ν) which forces γ and ν to be equal. ■