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

In this paper, we fix N -many l$\text{}_{2}$-Hilbert spaces Hk whose dimensions are $n_{k} \in \mathbb{N}^{\infty} = \mathbb{N} \cup \{\infty\}$ for $k = 1, \ldots, N \in \mathbb{N} \backslash \{1\}$. 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 $\bigcup_{k=1}^{N} \mathcal{B}_{k}$, where Bk are the orthonormal bases of $H_{k}$ , for k = 1, . . . , N.