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In this chapter, we study two different types of \(\mathbb {R}\) -linear operators acting on a system \(\mathcal {H}\) of all scaled hypercomplex numbers of \(\underset {t\in \mathbb {R}}{\sqcup }\mathbb {H}_{t}\) , which forms an algebra over \(\mathbb {R}\) (in short, a \(\mathbb {R}\) -algebra), where \(\mathbb {H}_{t}\) are the \(\mathbb {R}\) -algebras of t-scaled hypercomplex numbers for all scales t from the real numbers of \(\mathbb {R}\) . In particular, we are interested in (i) how every continuous function \(g\in \mathcal {C}\) on \(\mathbb {R}\) acts on \(\mathcal {H}\) as a certain shift operator, and (ii) how each semigroup element \(\left (s,\left (a,b\right )\right )\in \left (\mathbb {R}\times \mathbb {C}^{2},\boxplus \right )\) acts on \(\mathcal {H}\) as a multiplication operator.

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Linear Operators on a System \(\mathcal {H}\) of All Scaled Hypercomplex Numbers over \(\mathbb {R}\)

  • 会议论文

  • 发表日期：

    2026年

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  • Ilwoo Cho

摘要

In this chapter, we study two different types of \(\mathbb {R}\) -linear operators acting on a system \(\mathcal {H}\) of all scaled hypercomplex numbers of \(\underset {t\in \mathbb {R}}{\sqcup }\mathbb {H}_{t}\) , which forms an algebra over \(\mathbb {R}\) (in short, a \(\mathbb {R}\) -algebra), where \(\mathbb {H}_{t}\) are the \(\mathbb {R}\) -algebras of t-scaled hypercomplex numbers for all scales t from the real numbers of \(\mathbb {R}\) . In particular, we are interested in (i) how every continuous function \(g\in \mathcal {C}\) on \(\mathbb {R}\) acts on \(\mathcal {H}\) as a certain shift operator, and (ii) how each semigroup element \(\left (s,\left (a,b\right )\right )\in \left (\mathbb {R}\times \mathbb {C}^{2},\boxplus \right )\) acts on \(\mathcal {H}\) as a multiplication operator.