Semi-weak $k$-hyponormality of recursively generated weighted shifts with first two equal weights

Abstract

Semi-weak $k$-hyponormality has been considered to study the weak subnormality of Hilbert space operators. In this paper, we consider a recursive weight sequence $\alpha(a,b,\rho)$ induced by two atomic Berger measure with atoms $\{a,b\}$ and density $\rho$ for $0 < a,b,\rho <1$, and the corresponding weighted shift $W_{\alpha(a,b,\rho)}$. For all $k \ge 2$, we characterize semi-weak $k$-hyponormalities of recursively generated weighted shifts with first two equal weights. We also show that a semi-weakly $k$-hyponormal weighted shift needs not satisfy the flatness property, in which equality of first two weights forces all weights to be equal.