Triangular projection on $\boldsymbol{S}_p,~0<p<1$, as $\boldsymbol{p}$ approaches 1
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| Format: | Preprint |
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2024
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| _version_ | 1866916123268612096 |
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| author | Aleksandrov, A. B. Peller, V. V. |
| author_facet | Aleksandrov, A. B. Peller, V. V. |
| contents | This is a continuation of our recent paper. We continue studying properties of the triangular projection ${\mathscr P}_n$ on the space of $n\times n$ matrices. We establish sharp estimates for the $p$-norms of ${\mathscr P}_n$ as an operator on the Schatten--von Neumann class $\boldsymbol{S}_p$ for $0<p<1$. Our estimates are uniform in $n$ and $p$ as soon as $p$ is separated away from 0. The main result of the paper shows that for $p\in(0,1)$, the $p$-norms of ${\mathscr P}_n$ on $\boldsymbol{S}_p$ behave as $n\to\infty$ and $p\to1$ as $n^{1/p-1}\min\big\{(1-p)^{-1},\log n\big\}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_08045 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Triangular projection on $\boldsymbol{S}_p,~0<p<1$, as $\boldsymbol{p}$ approaches 1 Aleksandrov, A. B. Peller, V. V. Functional Analysis Classical Analysis and ODEs Complex Variables 47B10 This is a continuation of our recent paper. We continue studying properties of the triangular projection ${\mathscr P}_n$ on the space of $n\times n$ matrices. We establish sharp estimates for the $p$-norms of ${\mathscr P}_n$ as an operator on the Schatten--von Neumann class $\boldsymbol{S}_p$ for $0<p<1$. Our estimates are uniform in $n$ and $p$ as soon as $p$ is separated away from 0. The main result of the paper shows that for $p\in(0,1)$, the $p$-norms of ${\mathscr P}_n$ on $\boldsymbol{S}_p$ behave as $n\to\infty$ and $p\to1$ as $n^{1/p-1}\min\big\{(1-p)^{-1},\log n\big\}$. |
| title | Triangular projection on $\boldsymbol{S}_p,~0<p<1$, as $\boldsymbol{p}$ approaches 1 |
| topic | Functional Analysis Classical Analysis and ODEs Complex Variables 47B10 |
| url | https://arxiv.org/abs/2402.08045 |