Journal article
Semi-hyponormality of commuting pairs of Hilbert space operators
Bulletin des sciences mathématiques, Vol.205, 103718
12/2025
DOI: 10.1016/j.bulsci.2025.103718
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(Z+2). 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.
Details
- Title: Subtitle
- Semi-hyponormality of commuting pairs of Hilbert space operators
- Creators
- Raúl E. Curto - Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USAJasang Yoon - The University of Texas Rio Grande Valley
- Resource Type
- Journal article
- Publication Details
- Bulletin des sciences mathématiques, Vol.205, 103718
- DOI
- 10.1016/j.bulsci.2025.103718
- ISSN
- 0007-4497
- eISSN
- 1952-4773
- Publisher
- Elsevier Masson SAS
- Grant note
- DMS-2247167 / National Science Foundation (https://doi.org/10.13039/100000001)
- Language
- English
- Electronic publication date
- 08/28/2025
- Date published
- 12/2025
- Academic Unit
- Mathematics
- Record Identifier
- 9984958286402771
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