Frames generated by the functional calculus and function frames of a normal operator
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866912891750318080 |
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| author | Idrissi, Nizar El |
| author_facet | Idrissi, Nizar El |
| contents | In this article, we prove that sequences generated by the functional calculus $(f(T)(e_n))_{n \in \mathbb{N}}$ can be equivalently written as function sequences $(f_n(T) g)_{n \in \mathbb{N}}$, when $T$ is normal and $g$ a cyclic vector for $T$. Here, $(e_n)_{n \in \mathbb{N}}$ is a sequence of vectors, $T$ is a bounded normal operator, $f$ and $(f_n)_{n \in \mathbb{N}}$ are functions defined on a neighborhood of the spectrum $σ(T)$, and $g$ is a cyclic vector for $T$. After that, we characterize the frame property of such sequences in terms of the approximate point spectrum of $T^*$. Examples include certain operators (normal operators, compact operators, unilateral shifts, multiplication operators on Hardy spaces, etc.) that either generate only Riesz bases or allow redundancy. Our bridge theorem makes explicit the structural equivalence between frames generated by the functional calculus and function frames. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_10290 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Frames generated by the functional calculus and function frames of a normal operator Idrissi, Nizar El Functional Analysis 42C15, 47B02, 47A60, 47A10, 46C05 In this article, we prove that sequences generated by the functional calculus $(f(T)(e_n))_{n \in \mathbb{N}}$ can be equivalently written as function sequences $(f_n(T) g)_{n \in \mathbb{N}}$, when $T$ is normal and $g$ a cyclic vector for $T$. Here, $(e_n)_{n \in \mathbb{N}}$ is a sequence of vectors, $T$ is a bounded normal operator, $f$ and $(f_n)_{n \in \mathbb{N}}$ are functions defined on a neighborhood of the spectrum $σ(T)$, and $g$ is a cyclic vector for $T$. After that, we characterize the frame property of such sequences in terms of the approximate point spectrum of $T^*$. Examples include certain operators (normal operators, compact operators, unilateral shifts, multiplication operators on Hardy spaces, etc.) that either generate only Riesz bases or allow redundancy. Our bridge theorem makes explicit the structural equivalence between frames generated by the functional calculus and function frames. |
| title | Frames generated by the functional calculus and function frames of a normal operator |
| topic | Functional Analysis 42C15, 47B02, 47A60, 47A10, 46C05 |
| url | https://arxiv.org/abs/2107.10290 |