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

TL;DR

This paper establishes a novel connection between weighted shift operators' moment infinite divisibility and conditional positive definiteness, providing new insights into the relationship between k-hyponormality and n-contractivity.

Contribution

It introduces a characterization of moment infinite divisibility via CPD matrices and links it to the interaction between k-hyponormality and n-contractivity in weighted shifts.

Findings

W_{ ext{α}} is ext{MID} iff certain log Hankel matrices are CPD.

Contractive weighted shifts are ext{MID} iff all their moment matrices' powers are CPD.

New theoretical bridge between hyponormality and contractivity in operator theory.

Abstract

For sequences $\alpha \equiv \{\alpha_n\}_{n=0}^{\infty}$ of positive real numbers, called weights, we study the weighted shift operators $W_{\alpha}$ having the property of moment infinite divisibility ($\mathcal{MID}$); that is, for any $p > 0$, the Schur power $W_{\alpha}^p$ is subnormal. We first prove that $W_{\alpha}$ is $\mathcal{MID}$ if and only if certain infinite matrices $\log M_{\gamma}(0)$ and $\log M_{\gamma}(1)$ are conditionally positive definite (CPD). Here $\gamma$ is the sequence of moments associated with $\alpha$, $M_{\gamma}(0),M_{\gamma}(1)$ are the canonical Hankel matrices whose positive semi-definiteness determines the subnormality of $W_{\alpha}$, 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 $W_{\alpha}$ is $\mathcal{MID}$ if and only if for all $p>0$, $M_{\gamma}^p(0)$ and $M_{\gamma}^p(1)$ are CPD.