TY  - JOUR
ID  - Alpay2023
LB  - Alpay2023
AU  - Alpay, Daniel
AU  - Cho, Ilwoo
TI  - Operators induced by certain hypercomplex systems
ST  - Operators induced by certain hypercomplex systems
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2023
VL  - 43
IS  - 3
SP  - 275
EP  - 333
DO  - https://doi.org/10.7494/OpMath.2023.43.3.275
UR  - https://doi.org/10.7494/OpMath.2023.43.3.275
AB  - 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.
KW  - scaled hypercomplex ring
KW  - scaled hypercomplex monoids
KW  - representations
KW  - scaled-spectral forms
KW  - scaled-spectralization
SN  - 1232-9274
ER  -