The Schur multiplier norm and its dual norm
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915956665614336 |
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| author | Christensen, Erik |
| author_facet | Christensen, Erik |
| contents | We present a formula for the Schur multiplier norm of a complex self-adjoint matrix, and a formula for the norm, which is dual to the Schur multiplier norm, of a self-adjoint matrix.
For a complex self-adjoint $n \times n $ matrix $X$ we show that its Schur multiplier norm is determined by $$ \|X\|_S = \min \{\, \|\mathrm{diag}(P)\|_\infty \, :\, - P \leq X \leq P \, \}.$$
The dual space of $( M_n(\bc), \|.\|_S)$ is $(M_n(\bc), \|.\|_{cbB}).$ For $X=X^*:$ $$ \|X\|_{cbB} = \min \{ \, \mathrm{Tr}_n\big(Δ(λ)\big)\, :\, λ\in \br^n, \, - Δ(λ) \leq X \leq Δ(λ)\,\}. $$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_24670 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Schur multiplier norm and its dual norm Christensen, Erik Functional Analysis Operator Algebras 15A39, 15A45, 15A60, 46L07, 47A30, 81P47 We present a formula for the Schur multiplier norm of a complex self-adjoint matrix, and a formula for the norm, which is dual to the Schur multiplier norm, of a self-adjoint matrix. For a complex self-adjoint $n \times n $ matrix $X$ we show that its Schur multiplier norm is determined by $$ \|X\|_S = \min \{\, \|\mathrm{diag}(P)\|_\infty \, :\, - P \leq X \leq P \, \}.$$ The dual space of $( M_n(\bc), \|.\|_S)$ is $(M_n(\bc), \|.\|_{cbB}).$ For $X=X^*:$ $$ \|X\|_{cbB} = \min \{ \, \mathrm{Tr}_n\big(Δ(λ)\big)\, :\, λ\in \br^n, \, - Δ(λ) \leq X \leq Δ(λ)\,\}. $$ |
| title | The Schur multiplier norm and its dual norm |
| topic | Functional Analysis Operator Algebras 15A39, 15A45, 15A60, 46L07, 47A30, 81P47 |
| url | https://arxiv.org/abs/2505.24670 |