Spectral properties of certain operators on the free Hilbert space \mathfrak{F}[H_{1},...,H_{N}] and the semicircular law
In this paper, we fix \(N\)-many \(l^2\)-Hilbert spaces \(H_k\) whose dimensions are \(n_{k} \in \mathbb{N}^{\infty}=\mathbb{N} \cup \{\infty\}\), for \(k=1,\ldots,N\), for \(N \in \mathbb{N}\setminus\{1\}\). And then, construct a Hilbert space \(\mathfrak{F}=\mathfrak{F}[H_{1},\ldots,H_{N}]\) induc...
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AGH Univeristy of Science and Technology Press
2021
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| Acceso en línea: | https://doi.org/10.7494/OpMath.2021.41.6.755 https://doaj.org/article/17b587ffd90944e5907aee5bf60351e5 |
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oai:doaj.org-article:17b587ffd90944e5907aee5bf60351e52021-11-29T22:51:48ZSpectral properties of certain operators on the free Hilbert space \mathfrak{F}[H_{1},...,H_{N}] and the semicircular law1232-9274https://doi.org/10.7494/OpMath.2021.41.6.755https://doaj.org/article/17b587ffd90944e5907aee5bf60351e52021-11-01T00:00:00Zhttps://www.opuscula.agh.edu.pl/vol41/6/art/opuscula_math_4137.pdfhttps://doaj.org/toc/1232-9274In this paper, we fix \(N\)-many \(l^2\)-Hilbert spaces \(H_k\) whose dimensions are \(n_{k} \in \mathbb{N}^{\infty}=\mathbb{N} \cup \{\infty\}\), for \(k=1,\ldots,N\), for \(N \in \mathbb{N}\setminus\{1\}\). And then, construct a Hilbert space \(\mathfrak{F}=\mathfrak{F}[H_{1},\ldots,H_{N}]\) induced by \(H_{1},\ldots,H_{N}\), and study certain types of operators on \(\mathfrak{F}\). In particular, we are interested in so-called jump-shift operators. The main results (i) characterize the spectral properties of these operators, and (ii) show how such operators affect the semicircular law induced by \(\bigcup^N_{k=1} \mathcal{B}_{k}\), where \(\mathcal{B}_{k}\) are the orthonormal bases of \(H_{k}\), for \(k=1,\ldots,N\).Ilwoo ChoAGH Univeristy of Science and Technology Pressarticleseparable hilbert spacesfree hilbert spacesjump operatorsshift operatorsjump-shift operatorssemicircular elementsApplied mathematics. Quantitative methodsT57-57.97ENOpuscula Mathematica, Vol 41, Iss 6, Pp 755-803 (2021) |
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DOAJ |
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DOAJ |
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EN |
| topic |
separable hilbert spaces free hilbert spaces jump operators shift operators jump-shift operators semicircular elements Applied mathematics. Quantitative methods T57-57.97 |
| spellingShingle |
separable hilbert spaces free hilbert spaces jump operators shift operators jump-shift operators semicircular elements Applied mathematics. Quantitative methods T57-57.97 Ilwoo Cho Spectral properties of certain operators on the free Hilbert space \mathfrak{F}[H_{1},...,H_{N}] and the semicircular law |
| description |
In this paper, we fix \(N\)-many \(l^2\)-Hilbert spaces \(H_k\) whose dimensions are \(n_{k} \in \mathbb{N}^{\infty}=\mathbb{N} \cup \{\infty\}\), for \(k=1,\ldots,N\), for \(N \in \mathbb{N}\setminus\{1\}\). And then, construct a Hilbert space \(\mathfrak{F}=\mathfrak{F}[H_{1},\ldots,H_{N}]\) induced by \(H_{1},\ldots,H_{N}\), and study certain types of operators on \(\mathfrak{F}\). In particular, we are interested in so-called jump-shift operators. The main results (i) characterize the spectral properties of these operators, and (ii) show how such operators affect the semicircular law induced by \(\bigcup^N_{k=1} \mathcal{B}_{k}\), where \(\mathcal{B}_{k}\) are the orthonormal bases of \(H_{k}\), for \(k=1,\ldots,N\). |
| format |
article |
| author |
Ilwoo Cho |
| author_facet |
Ilwoo Cho |
| author_sort |
Ilwoo Cho |
| title |
Spectral properties of certain operators on the free Hilbert space \mathfrak{F}[H_{1},...,H_{N}] and the semicircular law |
| title_short |
Spectral properties of certain operators on the free Hilbert space \mathfrak{F}[H_{1},...,H_{N}] and the semicircular law |
| title_full |
Spectral properties of certain operators on the free Hilbert space \mathfrak{F}[H_{1},...,H_{N}] and the semicircular law |
| title_fullStr |
Spectral properties of certain operators on the free Hilbert space \mathfrak{F}[H_{1},...,H_{N}] and the semicircular law |
| title_full_unstemmed |
Spectral properties of certain operators on the free Hilbert space \mathfrak{F}[H_{1},...,H_{N}] and the semicircular law |
| title_sort |
spectral properties of certain operators on the free hilbert space \mathfrak{f}[h_{1},...,h_{n}] and the semicircular law |
| publisher |
AGH Univeristy of Science and Technology Press |
| publishDate |
2021 |
| url |
https://doi.org/10.7494/OpMath.2021.41.6.755 https://doaj.org/article/17b587ffd90944e5907aee5bf60351e5 |
| work_keys_str_mv |
AT ilwoocho spectralpropertiesofcertainoperatorsonthefreehilbertspacemathfrakfh1hnandthesemicircularlaw |
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1718406847189221376 |