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

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