Opuscula Math. 41, no. 6 (2021), 755-803

Opuscula Mathematica

Spectral properties of certain operators on the free Hilbert space \(\mathfrak{F}[H_{1},...,H_{N}]\) and the semicircular law

Ilwoo Cho

Abstract. 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 \(k=1,\ldots,N\).

Keywords: separable Hilbert spaces, free Hilbert spaces, jump operators, shift operators, jump-shift operators, semicircular elements.

Mathematics Subject Classification: 46L10, 46L54, 47L30, 47L55.

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  • Ilwoo Cho
  • St. Ambrose University, Department of Mathematics and Statistics, 518 W. Locust St., Davenport, Iowa, 52803, U.S.A.
  • Communicated by P.A. Cojuhari.
  • Received: 2020-11-05.
  • Accepted: 2021-04-03.
  • Published online: 2021-11-29.
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Cite this article as:
Ilwoo Cho, Spectral properties of certain operators on the free Hilbert space \(\mathfrak{F}[H_{1},...,H_{N}]\) and the semicircular law, Opuscula Math. 41, no. 6 (2021), 755-803, https://doi.org/10.7494/OpMath.2021.41.6.755

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