Bulletin of the
Korean Mathematical Society BKMS

Let $m\in \mathbb{N}_{0}$, $p>1$ and $$\alpha ^{\left[ m,p\right] }\left( x\right) :\sqrt{x},\left\{ \sqrt{\frac{\left( m+n-1\right) p-\left( m+n-2\right) }{\left( m+n\right) p-\left( m+n-1\right) }}\right\} _{n=1}^{\infty }.$$ In this paper, we consider the backward extensions of Bergman-type weighted shift $W_{\alpha ^{\left[ m,p\right] }\left( x\right) }.$ We consider its subnormality, $k$-hyponormality and positive quadratic hyponormality. Our results include all the results on Bergman weighted shift $W_{\alpha \left( x\right) }$ with $m\in \mathbb{N}$ and $$\alpha \left( x\right) :\sqrt{x},\sqrt{\frac{m}{m+1}},\sqrt{\frac{m+1}{m+2}},\sqrt{\frac{ m+2}{m+3}},\ldots .$$

Keywords: Bergman-type, subnormal, $k$-hyponormal, positively quadratically hyponormal, weighted shift

MSC numbers: 47B37, 47B20