Clarkson--McCarthy type inequalities, part I: $\ell_p$--$\ell_p$ and $\ell_q$--$\ell_p$ Schatten $p$-estimates
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912985802342400 |
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| author | Zhang, Teng |
| author_facet | Zhang, Teng |
| contents | We characterize the matrices $U=(u_{ij})$ for which the operator square-sum identity $$\sum_{i=1}^m\Big|\sum_{j=1}^n u_{ij}A_j\Big|^2=\sum_{j=1}^n|A_j|^2$$ holds for all Schatten-class operators $A_1,\ldots,A_n$; this happens exactly when $U$ is an isometry.Using this characterization, we establish Clarkson--McCarthy type inequalities for several classes of operator families, including $\ell_p$--$\ell_p$ estimates and mixed $\ell_q$--$\ell_p$ estimates.We also obtain a multivariable extension of the Ball--Carlen--Lieb $2$-uniform convexity inequality and a weaker bound toward Audenaert's norm-compression conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_21961 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Clarkson--McCarthy type inequalities, part I: $\ell_p$--$\ell_p$ and $\ell_q$--$\ell_p$ Schatten $p$-estimates Zhang, Teng Functional Analysis 47A30, 46B20, 15A60 We characterize the matrices $U=(u_{ij})$ for which the operator square-sum identity $$\sum_{i=1}^m\Big|\sum_{j=1}^n u_{ij}A_j\Big|^2=\sum_{j=1}^n|A_j|^2$$ holds for all Schatten-class operators $A_1,\ldots,A_n$; this happens exactly when $U$ is an isometry.Using this characterization, we establish Clarkson--McCarthy type inequalities for several classes of operator families, including $\ell_p$--$\ell_p$ estimates and mixed $\ell_q$--$\ell_p$ estimates.We also obtain a multivariable extension of the Ball--Carlen--Lieb $2$-uniform convexity inequality and a weaker bound toward Audenaert's norm-compression conjecture. |
| title | Clarkson--McCarthy type inequalities, part I: $\ell_p$--$\ell_p$ and $\ell_q$--$\ell_p$ Schatten $p$-estimates |
| topic | Functional Analysis 47A30, 46B20, 15A60 |
| url | https://arxiv.org/abs/2410.21961 |