Riesz Bases in Krein Spaces
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912158941446144 |
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| author | Jahan, Shah Johnson, P. Sam |
| author_facet | Jahan, Shah Johnson, P. Sam |
| contents | We start by introducing and studying the definition of a Riesz basis in a Krein space $(\mathcal{K},[.,.])$, along with a condition under which a Riesz basis becomes a Bessel sequence. The concept of biorthogonal sequence in Krein spaces is also introduced, providing an equivalent characterization of a Riesz basis. Additionally, we explore the concept of the Gram matrix, defined as the sum of a positive and a negative Gram matrices, and specify conditions under which the Gram matrix becomes bounded in Krein spaces. Further, we characterize the conditions under which the Gram matrices $\{[f_n,f_j]_{n,j \in I_+}\}$ and $\{[f_n,f_j]_{n,j \in I_-}\}$ become bounded invertible operators. Finally, we provide an equivalent characterization of a Riesz basis in terms of Gram matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11935 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Riesz Bases in Krein Spaces Jahan, Shah Johnson, P. Sam Functional Analysis 42C15, 46C05, 46C20 We start by introducing and studying the definition of a Riesz basis in a Krein space $(\mathcal{K},[.,.])$, along with a condition under which a Riesz basis becomes a Bessel sequence. The concept of biorthogonal sequence in Krein spaces is also introduced, providing an equivalent characterization of a Riesz basis. Additionally, we explore the concept of the Gram matrix, defined as the sum of a positive and a negative Gram matrices, and specify conditions under which the Gram matrix becomes bounded in Krein spaces. Further, we characterize the conditions under which the Gram matrices $\{[f_n,f_j]_{n,j \in I_+}\}$ and $\{[f_n,f_j]_{n,j \in I_-}\}$ become bounded invertible operators. Finally, we provide an equivalent characterization of a Riesz basis in terms of Gram matrices. |
| title | Riesz Bases in Krein Spaces |
| topic | Functional Analysis 42C15, 46C05, 46C20 |
| url | https://arxiv.org/abs/2412.11935 |