#### Semihyponormality of Commuting Pairs of Hilbert Space Operators (continued)

**Professor Raúl Curto; Department of Mathematics, University of Iowa**

We will first describe an explicit formula for the square root of positive 2 x 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_(α, β) (a, x, y) of 2-variable weighted shifts, we have been able to completely identify the parametric regions in the open unit cube where W_(α, β) (a, x, y) is subnormal, hyponormal, semi- hyponormal, and weakly hyponormal. As a result, we describe in detail concrete sub-regions where, for instance, weak hyponormality holds but semi- hyponormality does not hold. To accomplish this, we employ a new technique emanating from the homogeneous orthogonal decomposition of l²(Z²+). The talk is based on joint work with Jasang Yoon (The University of Texas Rio Grande Valley).We will first describe an explicit formula for the square root of positive 2 x 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_(α, β) (a, x, y) of 2-variable weighted shifts, we have been able to completely identify the parametric regions in the open unit cube where W_(α, β) (a, x, y) is subnormal, hyponormal, semi- hyponormal, and weakly hyponormal. As a result, we describe in detail concrete sub-regions where, for instance, weak hyponormality holds but semi- hyponormality does not hold. To accomplish this, we employ a new technique emanating from the homogeneous orthogonal decomposition of l²(Z²+). The talk is based on joint work with Jasang Yoon (The University of Texas Rio Grande Valley).

Those who wish to participate remotely may do so via Zoom at https://uiowa.zoom.us/j/95316149275