# Quasi-analytic Vectors and Hamburger Moment Problem (Operator Theory)

Analytic and quasi-analytic vectorsGuided 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\sum_{n=0}^{\infty}\lVert{A^n x}\rVert\f...