Weighted shifts relevant to CPD matrices and their examples

Abstract

In 1990 R. Curto introduced the notion of n-hyponormality which provides a bridge between subnormal and hyponormal operators. The study of n-hyponormal weighted shifts has been well developed by several mathematicians. In this paper we introduce a property $CP\left(n\right)$ for weighted shifts related to $(n+1)\times (n+1)$ conditionally positive definite matrices, which generalizes n-hyponormality for weighted shifts. First the flatness of a weighted shift with properties $CP\left(2\right)$ and $CP\left(3\right)$ is considered, yielding a result which generalizes previous work. A formula for property $CP\left(n\right)$ is constructed, which distinguishes the classes of weighted shifts with property $CP\left(n\right)$. We introduce an algorithm to construct weighted shifts with property $CP\left(n\right)$ and detect the structure related to property $CP\left(n\right)$. Finally, we discuss property $CP\left(n\right)$ of a homographic-type weighted shift with a constraint condition.