A Moment Problem and Joint q-Isometry Tuples

Abstract

The following moment problem arises in the classification of joint q-isometry multi-shifts which admit jointly subnormal spherical Cauchy dual tuples: For a polynomial $p : [0, \infty ) \rightarrow (0, \infty )$ such that $p(0)=1$, find a probability measure $\mu$ on [0, 1] such that

$$\begin{aligned} \frac{1}{p(l)}=\int _{[0, 1]}t^ld\mu (t)~(l \in {\mathbb {N}}). \end{aligned}$$

In an attempt toward solution of this problem, we draw upon an interesting connection between the moment problem and Hermite interpolation which we further employ to find explicit expression for the representing measure $\mu$ whenever the polynomial p is reducible over ${\mathbb {R}}$. This description is then used to compute Berger measures of spherical Cauchy dual tuples of a family of joint q-isometry multi-shifts, which in particular includes Drury–Arveson and Dirichlet d-shifts. Finally, we discuss several examples where the polynomial p is not reducible over ${\mathbb {R}}$ and highlight some difficulties in the study of corresponding moment problem.