Rescaling of unconditional Schauder frames in Hilbert spaces and completely bounded maps
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909721786580992 |
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| author | Tselishchev, Anton |
| author_facet | Tselishchev, Anton |
| contents | We prove that if every element $u$ in a Hilbert space $H$ admits a representation as unconditionally convergent series $$u=\sum_{k=1}^\infty \langle u, y_k\rangle x_k,$$ then there exist nonzero scalars $\{α_k\}_{k=1}^\infty$ such that both sequences $\{α_k x_k\}_{k=1}^\infty$ and $\{\overlineα_k^{-1}y_k\}_{k=1}^\infty$ are frames. Our result has the following equivalent reformulation: if $Φ:\ell^\infty\to B(H)$ is a bounded linear map such that for every element of the unit vector basis $e_k$ in $\ell^\infty$ the operator $Φ(e_k)$ has rank one, then $Φ$ is completely bounded. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_02802 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rescaling of unconditional Schauder frames in Hilbert spaces and completely bounded maps Tselishchev, Anton Functional Analysis Operator Algebras 46B15, 46L07 We prove that if every element $u$ in a Hilbert space $H$ admits a representation as unconditionally convergent series $$u=\sum_{k=1}^\infty \langle u, y_k\rangle x_k,$$ then there exist nonzero scalars $\{α_k\}_{k=1}^\infty$ such that both sequences $\{α_k x_k\}_{k=1}^\infty$ and $\{\overlineα_k^{-1}y_k\}_{k=1}^\infty$ are frames. Our result has the following equivalent reformulation: if $Φ:\ell^\infty\to B(H)$ is a bounded linear map such that for every element of the unit vector basis $e_k$ in $\ell^\infty$ the operator $Φ(e_k)$ has rank one, then $Φ$ is completely bounded. |
| title | Rescaling of unconditional Schauder frames in Hilbert spaces and completely bounded maps |
| topic | Functional Analysis Operator Algebras 46B15, 46L07 |
| url | https://arxiv.org/abs/2508.02802 |