Fixed points of completely positive maps and their dual maps
Abstract Let A ⊂ B ( H ) $\mathcal {A} \subset{\mathcal {B}}(\mathcal {H})$ be a row contraction and Φ A $\Phi _{\mathcal {A}}$ determined by A $\mathcal {A}$ be a completely positive map on B ( H ) ${\mathcal {B}}(\mathcal {H})$ . In this paper, we mainly consider fixed points of Φ A $\Phi _{\mathc...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2022-12-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s13660-022-02903-z |
| _version_ | 1797973404202041344 |
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| author | Haiyan Zhang Yanni Dou |
| author_facet | Haiyan Zhang Yanni Dou |
| author_sort | Haiyan Zhang |
| collection | DOAJ |
| description | Abstract Let A ⊂ B ( H ) $\mathcal {A} \subset{\mathcal {B}}(\mathcal {H})$ be a row contraction and Φ A $\Phi _{\mathcal {A}}$ determined by A $\mathcal {A}$ be a completely positive map on B ( H ) ${\mathcal {B}}(\mathcal {H})$ . In this paper, we mainly consider fixed points of Φ A $\Phi _{\mathcal {A}}$ and its dual map Φ A † $\Phi _{\mathcal {A}}^{\dagger}$ . It is given that Φ A ( X ) ≤ X $\Phi _{\mathcal {A}}(X)\leq X $ (or Φ A ( X ) ≥ X $\Phi _{\mathcal {A}}(X)\geq X $ ) implies Φ A ( X ) = X $\Phi _{\mathcal {A}}(X)= X$ and Φ A † ( X ) = X $\Phi _{\mathcal {A}}^{\dagger}(X)= X$ when X ∈ B ( H ) $X\in {\mathcal {B}}(\mathcal {H})$ is a compact operator. Some necessary conditions of Φ A ( X ) = X $\Phi _{\mathcal {A}}(X)= X$ and Φ A † ( X ) = X $\Phi _{\mathcal {A}}^{\dagger}(X)= X$ are given. |
| first_indexed | 2024-04-11T04:03:44Z |
| format | Article |
| id | doaj.art-1a1240c649c0491eb68ede5569a041df |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-04-11T04:03:44Z |
| publishDate | 2022-12-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-1a1240c649c0491eb68ede5569a041df2023-01-01T12:30:06ZengSpringerOpenJournal of Inequalities and Applications1029-242X2022-12-01202211910.1186/s13660-022-02903-zFixed points of completely positive maps and their dual mapsHaiyan Zhang0Yanni Dou1School of Mathematics and Statistics, Shangqiu Normal UniversitySchool of Mathematics and Statistics, Shaanxi Normal UniversityAbstract Let A ⊂ B ( H ) $\mathcal {A} \subset{\mathcal {B}}(\mathcal {H})$ be a row contraction and Φ A $\Phi _{\mathcal {A}}$ determined by A $\mathcal {A}$ be a completely positive map on B ( H ) ${\mathcal {B}}(\mathcal {H})$ . In this paper, we mainly consider fixed points of Φ A $\Phi _{\mathcal {A}}$ and its dual map Φ A † $\Phi _{\mathcal {A}}^{\dagger}$ . It is given that Φ A ( X ) ≤ X $\Phi _{\mathcal {A}}(X)\leq X $ (or Φ A ( X ) ≥ X $\Phi _{\mathcal {A}}(X)\geq X $ ) implies Φ A ( X ) = X $\Phi _{\mathcal {A}}(X)= X$ and Φ A † ( X ) = X $\Phi _{\mathcal {A}}^{\dagger}(X)= X$ when X ∈ B ( H ) $X\in {\mathcal {B}}(\mathcal {H})$ is a compact operator. Some necessary conditions of Φ A ( X ) = X $\Phi _{\mathcal {A}}(X)= X$ and Φ A † ( X ) = X $\Phi _{\mathcal {A}}^{\dagger}(X)= X$ are given.https://doi.org/10.1186/s13660-022-02903-zQuantum operationDual operationFixed pointCompact operator |
| spellingShingle | Haiyan Zhang Yanni Dou Fixed points of completely positive maps and their dual maps Journal of Inequalities and Applications Quantum operation Dual operation Fixed point Compact operator |
| title | Fixed points of completely positive maps and their dual maps |
| title_full | Fixed points of completely positive maps and their dual maps |
| title_fullStr | Fixed points of completely positive maps and their dual maps |
| title_full_unstemmed | Fixed points of completely positive maps and their dual maps |
| title_short | Fixed points of completely positive maps and their dual maps |
| title_sort | fixed points of completely positive maps and their dual maps |
| topic | Quantum operation Dual operation Fixed point Compact operator |
| url | https://doi.org/10.1186/s13660-022-02903-z |
| work_keys_str_mv | AT haiyanzhang fixedpointsofcompletelypositivemapsandtheirdualmaps AT yannidou fixedpointsofcompletelypositivemapsandtheirdualmaps |