Hankel operators and the Stieltjes moment problem

Abstract

Let s be a non-vanishing Stieltjes moment sequence and let μ be a representing measure of it. We denote by μ n the image measure in C n of μ ⊗ σ n under the map ( t , ξ ) ↦ t ξ , where σ n is the rotation invariant probability measure on the unit sphere. We show that the closure of holomorphic polynomials in L 2 ( μ n ) is a reproducing kernel Hilbert space of analytic functions and describe various spectral properties of the corresponding Hankel operators with anti-holomorphic symbols. In particular, if n = 1 , we prove that there are nontrivial Hilbert–Schmidt Hankel operators with anti-holomorphic symbols if and only if s is exponentially bounded. In this case, the space of symbols of such operators is shown to be the classical Dirichlet space. We mention that the classical weighted Bergman spaces, the Hardy space and Fock type spaces fall in this setting.