Characterization of Stieltjes transforms of vector measures and an application to spectral theory
Let M and D respectively denote the formal operators of multiplication M: f(t)\(\to tf(t)\) and differentiation D:f\(\to f'\). The (formal) Widder differential operators \(L_ k\) are defined by \[ L_ k=c_ kM^{k- 1}D^{2k-1}M^ k,\quad k=1,2,...,where\quad c_ 1=1\quad and\quad c_ k=(-1)^ k[k!(k-2)!]^{-1}\quad for\quad k\geq 2. \] Let \(c_ 0\) denote the space of all continuous \({\mathbb{C}}\)-valued functions on [0,\(\infty)\) which vanish at infinity, equipped with the supremum norm. It is shown that a function f on (0,\(\infty)\) with values in a quasicomplete locally convex space X is the Stieltjes transform of an X-valued measure with domain the Borel subsets of [0,\(\infty)\), if and only if, f has weak derivatives of all orders and the linear operators \(\Phi_ k:c_ 0\to X\) defined by \[ \Phi_ k(g):\psi \to \int^{\infty}_{0}\psi (t)L_ k(g)(t)dt,\quad \psi \in c_ 0, \] for each \(k=1,2,...\), map the closed unit ball of \(c_ 0\) into a weakly compact subset of X (independent of k). This characterization is then used to establish a criterion, due to S. Kantorovitz in the case when X is a reflexive Banach space, for a continuous linear operator on X with spectrum in [0,\(\infty)\) to be a scalar-type spectral operator.
scientific article; zbMATH DE number 3923502