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

Abstract For sequences α ≡ { α n } n = 0 ∞ of positive real numbers, called weights, we study the weighted shift operators W α having the property of moment infinite divisibility ( MID ); that is, for any p > 0 , the Schur power W α p is subnormal. We first prove that W α is 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 MID if and only if for all p > 0 , M γ p ( 0 ) and M γ p ( 1 ) are CPD.