Opuscula Math. 43, no. 3 (2023), 275-333
https://doi.org/10.7494/OpMath.2023.43.3.275

 
Opuscula Mathematica

Operators induced by certain hypercomplex systems

Daniel Alpay
Ilwoo Cho

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.

Mathematics Subject Classification: 20G20, 46S10, 47S10.

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  • Communicated by P.A. Cojuhari.
  • Received: 2022-10-11.
  • Revised: 2023-04-14.
  • Accepted: 2023-04-17.
  • Published online: 2023-05-17.
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Cite this article as:
Daniel Alpay, Ilwoo Cho, Operators induced by certain hypercomplex systems, Opuscula Math. 43, no. 3 (2023), 275-333, https://doi.org/10.7494/OpMath.2023.43.3.275

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