TY  - JOUR
ID  - Cho2021
LB  - Cho2021
AU  - Cho, Ilwoo
TI  - Spectral properties of certain operators on the free Hilbert space \(\mathfrak{F}[H_{1},...,H_{N}]\) and the semicircular law
ST  - Spectral properties of certain operators on the free Hilbert space \(\mathfrak{F}[H_{1},...,H_{N}]\) and the semicircular law
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2021
VL  - 41
IS  - 6
SP  - 755
EP  - 803
DO  - https://doi.org/10.7494/OpMath.2021.41.6.755
UR  - https://doi.org/10.7494/OpMath.2021.41.6.755
AB  - 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 \(k=1,\ldots,N\).
KW  - separable Hilbert spaces
KW  - free Hilbert spaces
KW  - jump operators
KW  - shift operators
KW  - jump-shift operators
KW  - semicircular elements
SN  - 1232-9274
ER  -