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

Abstract

For sequences α≡{αn}n=0∞ of positive real numbers, called weights, we study the weighted shift operators Wα having the property of moment infinite divisibility (MID); that is, for any p>0, the Schur power Wαp is subnormal. We first prove that Wα is MID if and only if certain infinite matrices log⁡Mγ(0) and log⁡Mγ(1) are conditionally positive definite (CPD). Here γ is the sequence of moments associated with α, Mγ(0),Mγ(1) are the canonical Hankel matrices whose positive semi-definiteness determines the subnormality of Wα, 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 Wα is MID if and only if for all p>0, Mγp(0) and Mγp(1) are CPD.