@Article{Alpay2023,
author = "Alpay, Daniel and Cho, Ilwoo",
title = "Operators induced by certain hypercomplex systems",
journal = "Opuscula Math.",
year = "2023",
volume = "43",
number = "3",
pages = "275--333",
doi = "https://doi.org/10.7494/OpMath.2023.43.3.275",
url = "https://doi.org/10.7494/OpMath.2023.43.3.275",
abstract = "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.",
keywords = "scaled hypercomplex ring, scaled hypercomplex monoids, representations, scaled-spectral forms, scaled-spectralization",
issn = "1232-9274"
}