Is there any nontrivial compact generalized shift operator on Hilbert spaces?
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866911759832449024 |
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| author | Shirazi, Fatemah Ayatollah Zadeh Ebrahimifar, Fatemeh |
| author_facet | Shirazi, Fatemah Ayatollah Zadeh Ebrahimifar, Fatemeh |
| contents | In the following text for cardinal number $τ>0$, and self--map $φ:τ\toτ$ we show the generalized shift operator $σ_φ(\ell^2(τ))\subseteq\ell^2(τ)$ (where $σ_φ((x_α)_{α<τ})=(x_{φ(α)})_{α<τ}$ for $(x_α)_{α<τ}\in{\mathbb C}^τ$) if and only if $φ:τ\toτ$ is bounded and in this case $σ_φ\restriction_{\ell^2(τ)}:\ell^2(τ)\to\ell^2(τ)$ is continuous, consequently $σ_φ\restriction_{\ell^2(τ)}:\ell^2(τ)\to\ell^2(τ)$ is a compact operator if and only if $τ$ is finite. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1804_07921 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Is there any nontrivial compact generalized shift operator on Hilbert spaces? Shirazi, Fatemah Ayatollah Zadeh Ebrahimifar, Fatemeh Functional Analysis 46C99 In the following text for cardinal number $τ>0$, and self--map $φ:τ\toτ$ we show the generalized shift operator $σ_φ(\ell^2(τ))\subseteq\ell^2(τ)$ (where $σ_φ((x_α)_{α<τ})=(x_{φ(α)})_{α<τ}$ for $(x_α)_{α<τ}\in{\mathbb C}^τ$) if and only if $φ:τ\toτ$ is bounded and in this case $σ_φ\restriction_{\ell^2(τ)}:\ell^2(τ)\to\ell^2(τ)$ is continuous, consequently $σ_φ\restriction_{\ell^2(τ)}:\ell^2(τ)\to\ell^2(τ)$ is a compact operator if and only if $τ$ is finite. |
| title | Is there any nontrivial compact generalized shift operator on Hilbert spaces? |
| topic | Functional Analysis 46C99 |
| url | https://arxiv.org/abs/1804.07921 |