Conditional positive definiteness as a bridge between k-hyponormality and n-contractivity

Abstract

For sequences $α\equiv \{α_n\}_{n=0}^{\infty}$ of positive real numbers, called weights, we study the weighted shift operators $W_α$ having the property of moment infinite divisibility ($\mathcal{MID}$); that is, for any $p > 0$, the Schur power $W_α^p$ is subnormal. We first prove that $W_α$ is $\mathcal{MID}$ if and only if certain infinite matrices $\log M_γ(0)$ and $\log M_γ(1)$ are conditionally positive definite (CPD). Here $γ$ is the sequence of moments associated with $α$, $M_γ(0),M_γ(1)$ are the canonical Hankel matrices whose positive semi-definiteness determines the subnormality of $W_α$, and $\log$ is calculated entry-wise (i.e., in the sense of Schur or Hadamard). Next, we use conditional positive definiteness to establish a new bridge between $k$--hyponormality and $n$--contractivity, which sheds significant new light on how the two well known staircases from hyponormality to subnormality interact. As a consequence, we prove that a contractive weighted shift $W_α$ is $\mathcal{MID}$ if and only if for all $p>0$, $M_γ^p(0)$ and $M_γ^p(1)$ are CPD.