Cauchy Dual Subnormality for Toral 2-Isometric Operator-Valued 2-Variable Weighted Shift

In this paper, we show that if $\mathbf{T}=(T_1,T_2)$ is an analytic left-inverse commuting pair of toral $2$-isometries satisfying the joint kernel condition, then it is unitarily equivalent to an operator-valued weighted shift with invertible weights $\{W_I^{(j)}:j=1,2{\}}_{I\in {\mathbb{Z}}_{+}^2},$ where the initial weights $W_{0,0}^{(1)}$ and $W_{0,0}^{(2)}$ are positive operators. Moreover, if these initial weights commute, then the Cauchy dual ${\mathbf{T}}^{\mathrm{\prime }}:=(T_1^{\mathrm{\prime }},T_2^{\mathrm{\prime }})$ is jointly subnormal. We also construct an example in which the initial weights do not commute, and the corresponding Cauchy dual fails to be jointly subnormal.