Inner products on the Hilbert space $S_2$ of Hilbert--Schmidt operators
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913877232451584 |
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| author | Rios-Cangas, Josué I. |
| author_facet | Rios-Cangas, Josué I. |
| contents | This work presents a rigorous characterization of inner products on the Hilbert space $S_2$ of Hilbert--Schmidt operators. We first deal with a general setting of continuous sesquilinear forms on a Hilbert space $\mathcal H$, and provide a characterization of all inner products by means of positive operators in $\mathcal {B(H)}$. Next, we establish necessary and sufficient conditions for an operator in $\mathcal B(S_2)$ to be positive. Identifying an inner product with a positive operator enables us to rigorously describe inner products on $S_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_04541 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Inner products on the Hilbert space $S_2$ of Hilbert--Schmidt operators Rios-Cangas, Josué I. Functional Analysis Mathematical Physics Primary 46C50, Secondary 46C05, 47A65 This work presents a rigorous characterization of inner products on the Hilbert space $S_2$ of Hilbert--Schmidt operators. We first deal with a general setting of continuous sesquilinear forms on a Hilbert space $\mathcal H$, and provide a characterization of all inner products by means of positive operators in $\mathcal {B(H)}$. Next, we establish necessary and sufficient conditions for an operator in $\mathcal B(S_2)$ to be positive. Identifying an inner product with a positive operator enables us to rigorously describe inner products on $S_2$. |
| title | Inner products on the Hilbert space $S_2$ of Hilbert--Schmidt operators |
| topic | Functional Analysis Mathematical Physics Primary 46C50, Secondary 46C05, 47A65 |
| url | https://arxiv.org/abs/2506.04541 |