On indefinite-inner-product spaces induced by non-zero-scaled hypercomplex numbers

Abstract

In this paper, we consider a new type of adjoint $[⁎]$ on the algebra $H_t$ of all t-scaled hypercomplex numbers over the real field $R$, for all “non-zero” scales $t\in R\setminus \left\{0\right\}$. We show that such a $R$-adjoint $[⁎]$ generates a well-defined indefinite inner product ${[,]}_t$ on $H_t$, inducing a complete indefinite inner product space $H_t=(H_t,{[,]}_t)$ over $R$. Analysis and operator theory on $H_t$ is considered up to this adjoint $[⁎]$. As application, by regarding t-scaled hypercomplex numbers of $H_t$ as embedded subset $M_2^t$ of $M_2\left(C\right)$, the corresponding (usual operator-theoretic) spectral theory on $M_2^t$ is studied (over the complex field $C$). And we study relations between these usual spectral-theoretic results and the operator-theoretic results obtained from the $[⁎]$-depending structures; and then the free distributions of self-adjoint matrices of $M_2^t$ are characterized up to the normalized trace τ on $M_2\left(C\right)$.