Conditional positive definiteness as a bridge between \(k\)-hyponormality and \(n\)-contractivity (Q2032250)

In this paper, important connections between shift operators \(W_\alpha\) of weights \(\alpha\equiv\lbrace\alpha_{n}\rbrace_{n=0}^{\infty}\) having the property of moment infinite divisibility \((\mathcal{M}\mathcal{I}\mathcal{D})\), that is, for any \(p>0\), the Schur power \(W_{\alpha}^p\) is subnormal and infinite matrices \(\log M_{\gamma}(0)\) and \(\log M_{\gamma}(1)\) having the conditionally positiveness property (CPD) are given. In this context, \(\gamma\) is the sequence of moments associated with \(\alpha\), \(M_{\gamma}(0)\), \(M_{\gamma}(1)\) are the canonical Hankel matrices, \(\log\) is computed in the sense of Schur or Hadamard and CPD property for an infinite (scalar) matrix means that all of its principal minors of finite size \((a_{i,j})_{i,j=0}^k\) with \(a_{ji} = \bar{a}_{ij}\) for all \(i,j = 0, \dotsc ,k\), satisfy the condition \(\sum\limits_{i,j=0}^k a_{ij}\bar{c_{i}}c_{j}\geq 0\) for all complex vectors \(c\equiv(c_{0},c_{1},\dotsc,c_{k})^T\) such that \(c_{0}+c_{1}+ \dotsc +c_{k} = 0\). It is worth mentioning that the CPD property is used to established a new bridge between k-hyponormality and n-contractivity. As a consequence, it is proved that a contractive weighted shift \(W_\alpha\) is \(\mathcal{M}\mathcal{I}\mathcal{D}\) if and only if for all \(p>0\), \(M_{\gamma}^p(0)\), \(M_{\gamma}^p(1)\) are CPD.