Semi-hyponormality of commuting pairs of Hilbert space operators

We first find an explicit formula for the square root of positive $2 \times 2$ operator matrices with commuting entries, and then use it to define and study semi-hyponormality for commuting pairs of Hilbert space operators. \ For the well-known $3$--parameter family $W_{(\alpha,\beta)}(a,x,y)$ of $2$--variable weighted shifts, we completely identify the parametric regions in the open unit cube where $W_{(\alpha,\beta)}(a,x,y)$ is subnormal, hyponormal, semi-hyponormal, and weakly hyponormal. As a result, we describe in detail concrete sub-regions where each property holds. For instance, we identify the specific sub-region where weak hyponormality holds but semi-hyponormality does not hold, and vice versa. \ To accomplish this, we employ a new technique emanating from the homogeneous orthogonal decomposition of $\ell^2(\mathbb{Z}_+^2)$. The technique allows us to reduce the study of semi-hyponormality to positivity considerations of a sequence of $2 \times 2$ scalar matrices. It also requires a specific formula for the square root of $2 \times 2$ scalar and operator matrices, and we obtain that along the way. As an application of our main results, we show that the Drury-Arveson shift is {\it not} semi-hyponormal. Taken together, the new results offer a sharp contrast between the above-mentioned properties for unilateral weighted shifts and their $2$--variable counterparts.