The block matrix representations for the quasi-projection pairs on Hilbert $C^*$-modules
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917052564897792 |
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| author | Tian, Xiaoyi Xu, Qingxiang Fu, Chunhong |
| author_facet | Tian, Xiaoyi Xu, Qingxiang Fu, Chunhong |
| contents | A quasi-projection pair consists of two operators $P$ and $Q$ acting on a Hilbert $C^*$-module $H$, where $P$ is a projection and $Q$ is an idempotent satisfying $Q^*=(2P-I)Q(2P-I)$, in which $Q^*$ denotes the adjoint operator of $Q$, and $I$ is the identity operator on $H$. Such a pair is said to be harmonious if both $P(I-Q)$ and $(I-P)Q$ admit polar decompositions. The primary goal of this paper is to present the block matrix representations for a harmonious quasi-projection pair $(P,Q)$ on a Hilbert $C^*$-module, and additionally to derive new block matrix representations for the matched projection, the range projection, and the null space projection of $Q$. Several applications of these newly obtained block matrix representations are also explored. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_27068 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The block matrix representations for the quasi-projection pairs on Hilbert $C^*$-modules Tian, Xiaoyi Xu, Qingxiang Fu, Chunhong Functional Analysis Operator Algebras A quasi-projection pair consists of two operators $P$ and $Q$ acting on a Hilbert $C^*$-module $H$, where $P$ is a projection and $Q$ is an idempotent satisfying $Q^*=(2P-I)Q(2P-I)$, in which $Q^*$ denotes the adjoint operator of $Q$, and $I$ is the identity operator on $H$. Such a pair is said to be harmonious if both $P(I-Q)$ and $(I-P)Q$ admit polar decompositions. The primary goal of this paper is to present the block matrix representations for a harmonious quasi-projection pair $(P,Q)$ on a Hilbert $C^*$-module, and additionally to derive new block matrix representations for the matched projection, the range projection, and the null space projection of $Q$. Several applications of these newly obtained block matrix representations are also explored. |
| title | The block matrix representations for the quasi-projection pairs on Hilbert $C^*$-modules |
| topic | Functional Analysis Operator Algebras |
| url | https://arxiv.org/abs/2510.27068 |