School of Mathematical & Statistical Sciences Faculty Publications
Document Type
Article
Publication Date
12-2025
Abstract
We first find an explicit formula for the square root of positive 2 ×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 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 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 ℓ2(ℤ2+). The technique allows us to reduce the study of semi-hyponormality to positivity considerations of a sequence of 2 ×2 scalar matrices. It also requires a specific formula for the square root of 2 ×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 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.
Recommended Citation
Curto, Raúl E., and Jasang Yoon. "Semi-hyponormality of commuting pairs of Hilbert space operators." Bulletin des sciences mathématiques (2025): 103718. https://doi.org/10.1016/j.bulsci.2025.103718
Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.
Publication Title
Bulletin des Sciences Mathématiques
DOI
10.1016/j.bulsci.2025.103718
Comments
Original published version available at https://doi.org/10.1016/j.bulsci.2025.103718