Spectral properties of certain operators on the free Hilbert space F[H1,...,HN] and the semicircular law

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 $k=1,\ldots,N$.