Semi-hyponormality of commuting pairs of Hilbert space operators
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866914545621008384 |
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| author | Curto, Raul E. Yoon, Jasang |
| author_facet | Curto, Raul E. Yoon, Jasang |
| contents | 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 $W_{(α,β)}(a,x,y)$ 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 $\ell^2(\mathbb{Z}_+^2)$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_02197 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Semi-hyponormality of commuting pairs of Hilbert space operators Curto, Raul E. Yoon, Jasang Functional Analysis 47B20, 47B37, 47A13, 28A50 (Primary) 44A60, 47-04, 47A20 (Secondary) 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 $W_{(α,β)}(a,x,y)$ 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 $\ell^2(\mathbb{Z}_+^2)$. 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. |
| title | Semi-hyponormality of commuting pairs of Hilbert space operators |
| topic | Functional Analysis 47B20, 47B37, 47A13, 28A50 (Primary) 44A60, 47-04, 47A20 (Secondary) |
| url | https://arxiv.org/abs/2605.02197 |