Quasi-analytic Vectors and Hamburger Moment Problem (Operator Theory)
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 $x \in X$ is a $C^\infty$ vector if $x \in \bigcap_{n \geq 1}D(A^n)$. This is quite intuitive if we consider the differential operator. A vector is analytic if the series
has a positive radius of convergence. Finally, we say $x$ is quasi-analytic for $A$ provided that
or equivalently its nondecreasing majorant. Interestingly, if $A$ is symmetric, then $\lVert{A^nx}\rVert$ 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 $\mathscr{H}$. 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.