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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.12916 |
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| _version_ | 1866909080684068864 |
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| author | Abdelwahab, Hani Barreiro, Elisabete Calderón, Antonio J. Sánchez, José M. |
| author_facet | Abdelwahab, Hani Barreiro, Elisabete Calderón, Antonio J. Sánchez, José M. |
| contents | In this work we study a linear operator $f$ on a pre-euclidean space $\mathcal{V}$ by using properties of a corresponding graph. Given a basis $\B$ of $\mathcal{V}$, we present a decomposition of $\mathcal{V}$ as an orthogonal direct sum of certain linear subspaces $\{U_i\}_{i \in I}$, each one admitting a basis inherited from $\B$, in such way that $f = \sum_{i \in I}f_i$, being each $f_i$ a linear operator satisfying certain conditions respect with $U_i$. Considering new hypothesis, we assure the existence of an isomorphism between the graphs associated to $f$ relative to two different bases. We also study the minimality of $\mathcal{V}$ by using the graph associated to $f$ relative to $\B$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_12916 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Decompositions of linear operators on pre-euclidean spaces by means of graphs Abdelwahab, Hani Barreiro, Elisabete Calderón, Antonio J. Sánchez, José M. Representation Theory Functional Analysis In this work we study a linear operator $f$ on a pre-euclidean space $\mathcal{V}$ by using properties of a corresponding graph. Given a basis $\B$ of $\mathcal{V}$, we present a decomposition of $\mathcal{V}$ as an orthogonal direct sum of certain linear subspaces $\{U_i\}_{i \in I}$, each one admitting a basis inherited from $\B$, in such way that $f = \sum_{i \in I}f_i$, being each $f_i$ a linear operator satisfying certain conditions respect with $U_i$. Considering new hypothesis, we assure the existence of an isomorphism between the graphs associated to $f$ relative to two different bases. We also study the minimality of $\mathcal{V}$ by using the graph associated to $f$ relative to $\B$. |
| title | Decompositions of linear operators on pre-euclidean spaces by means of graphs |
| topic | Representation Theory Functional Analysis |
| url | https://arxiv.org/abs/2401.12916 |