Weakly $n$-hyponormal weighted shifts: a sufficient condition and their examples

Abstract

The $n$-hyponormal and weakly $n$-hyponormal weighted shifts were developed to study bridges of operators between the subnormal and hyponormal operators on an infinite dimensional complex Hilbert space about 30 years ago. In this paper we discuss the distinction between the classes of $n$-hyponormal and weakly $n$-hyponormal weighted shifts. For such a purpose we consider an arbitrary contractive hyponormal weighted shift $W_{\alpha }$ and find a sufficient condition for the weak $n$-hyponormality of $W_{\alpha }$. We provide a general technique for distinction between the $n$-hyponormality and the weak $n$-hyponormality of $W_{\alpha }$, and investigate the distinction between the classes of $n$-hyponormal and weakly $n$-hyponormal weighted shifts with Bergman shift and some other examples.