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Browsing by Subject "scaled hypercomplex ring"

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    Item type:Article, Access status: Open Access ,
    Operators induced by certain hypercomplex systems
    (Wydawnictwa AGH, 2023) Alpay, Daniel; Cho, Ilwoo
    In this paper, we consider a family $\{ \mathbb{H}_{t}\}_{t\in\mathbb{R}}$ of rings of hypercomplex numbers, indexed by the real numbers, which contain both the quaternions and the split-quaternions. We consider natural Hilbert-space representations $\{(\mathbb{C}^{2},\pi_{t})\}_{t\in\mathbb{R}}$ of the hypercomplex system $\{ \mathbb{H}_{t}\}_{t\in\mathbb{R}}$, and study the realizations $\pi_{t}(h)$ of hypercomplex numbers $h \in \mathbb{H}_{t}$, as $(2\times 2)$-matrices acting on $\mathbb{C}^{2}$, for an arbitrarily fixed scale $t \in \mathbb{R}$. Algebraic, operator-theoretic, spectral-analytic, and free-probabilistic properties of them are considered.