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Operators Induced By Certain Hypercomplex Systems, Daniel Alpay, Ilwoo Choo

Mathematics, Physics, and Computer Science Faculty Articles and Research

In this paper, we consider a family {H_t}_t∈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 {(C², π_t)}_t∈R of the hypercomplex system {H_t}_t∈R, and study the realizations π_t(h) of hypercomplex numbers h ∈ H_t, as (2 × 2)-matrices acting on C², for an arbitrarily fixed scale t ∈ R. Algebraic, operator-theoretic, spectral-analytic, and free-probabilistic properties of them are considered.