条件正定性作为k-次超正规性与n-合同性之间的桥梁
Conditional positive definiteness as a bridge between k-hyponormality and n-contractivity
AI总结：
研究权重序列诱导的加权移位算子的条件正定性，建立k-次超正规性与n-合同性之间的联系，揭示两者交互机制。
摘要：
对于正实数序列α≡{α_n}_{n=0}^∞（称作权重），我们研究具有动量无限可分性（MID）性质的加权移位算子W_α；即对于任何p>0，Schur幂W_α^p是次正规的。我们首先证明W_α是MID当且仅当某些无限矩阵logM_γ(0)和logM_γ(1)是条件正定的（CPD）。这里γ是与α相关的动量序列，M_γ(0),M_γ(1)是标准Hankel矩阵，其半正定性决定W_α的次正规性，log是逐项计算（即Schur或Hadamard意义）。接着，我们利用条件正定性建立k-次超正规性与n-合同性之间的新桥梁，揭示这两个已知的从超正规性到次正规性阶梯的交互机制。作为结果，我们证明一个合同的加权移位算子W_α是MID当且仅当对于所有p>0，M_γ^p(0)和M_γ^p(1)是CPD。
英文摘要：
For sequences $α\equiv \{α_n\}_{n=0}^{\infty}$ of positive real numbers, called weights, we study the weighted shift operators $W_α$ having the property of moment infinite divisibility ($\mathcal{MID}$); that is, for any $p > 0$, the Schur power $W_α^p$ is subnormal. We first prove that $W_α$ is $\mathcal{MID}$ 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 $\mathcal{MID}$ if and only if for all $p>0$, $M_γ^p(0)$ and $M_γ^p(1)$ are CPD.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
