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Semi-hyponormality of commuting pairs of Hilbert space operators
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Semi-hyponormality of commuting pairs of Hilbert space operators

Raul E Curto and Jasang Yoon
ArXiv.org
Cornell University
05/04/2026
DOI: 10.48550/arxiv.2605.02197
url
https://doi.org/10.48550/arxiv.2605.02197View
Preprint (Author's original) This preprint has not been evaluated by subject experts through peer review. Preprints may undergo extensive changes and/or become peer-reviewed journal articles. Open Access

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.
Mathematics - Functional Analysis

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