Hanner's Inequality For Positive Semidefinite Matrices
| Autor: | Chayes, Victoria M. |
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| Rok vydání: | 2021 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We prove an analogous Hanner's Inequality of $L^p$ spaces for positive semidefinite matrices. Let $||X||_p=\text{Tr}[(X^\ast X)^{p/2}]^{1/p}$ denote the $p$-Schatten norm of a matrix $X\in M_{n\times n}(\mathbb{C})$. We show that the inequality $||X+Y||_p^p+||X-Y||_p^p\geq (||X||_p+||Y||_p)^p+(|||X||_p-||Y||_p|)^p$ holds for $1\leq p\leq 2$ and reverses for $p\geq 2$ when $X,Y\in M_{n\times n}(\mathbb{C})^+$. This was previously known in the $1 |
| Databáze: | arXiv |
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