A simplified formula for the matched projection of an idempotent
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908699017084928 |
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| author | Xu, Qingxiang |
| author_facet | Xu, Qingxiang |
| contents | Let $\mathcal{L}(H)$ be the set of all adjointable operators on a Hilbert $C^*$-module $H$. For each $T\in\mathcal{L}(H)$, $T^*$ denotes its adjoint operator, and $|T^*|$ is the positive square root of $TT^*$. We establish a simplified formula for the matched projection $m(Q)$ of an idempotent $Q\in\mathcal{L}(H)$ as $$m(Q)=\frac{I+|Q^*|-|I-Q^*|}{2},$$ where $I$ is the identity operator on $H$. This explicit expression allows for the direct derivation of some basic properties of $m(Q)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_05970 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A simplified formula for the matched projection of an idempotent Xu, Qingxiang Functional Analysis Let $\mathcal{L}(H)$ be the set of all adjointable operators on a Hilbert $C^*$-module $H$. For each $T\in\mathcal{L}(H)$, $T^*$ denotes its adjoint operator, and $|T^*|$ is the positive square root of $TT^*$. We establish a simplified formula for the matched projection $m(Q)$ of an idempotent $Q\in\mathcal{L}(H)$ as $$m(Q)=\frac{I+|Q^*|-|I-Q^*|}{2},$$ where $I$ is the identity operator on $H$. This explicit expression allows for the direct derivation of some basic properties of $m(Q)$. |
| title | A simplified formula for the matched projection of an idempotent |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2512.05970 |