Certain operator algebras of star-like reducible $P\omega_n^*$ transformations

Abstract

Let $Z_n = 1, 2, 3, \ldots$ denote a distinct non-negative n-order collection of numbers, and $\alpha\omega_n^*$ denote a star-like transformation semigroup. The characterization of $P\omega_n^*$ star-like partial on the $\alpha\omega_n^*$ leads to the semigroup of linear operators. The research produced a completely new classical metamorphosis that was divided into inner product and norm parts. The study demonstrated that any specific star-like transformation $\lambda_i^*, \beta_j^* \in V^*$ is stable and uniformly continuous if there exists $T^{\vartheta^*}:(V^*, \left\langle v - \alpha^* u , u - \alpha^* v \right\rangle) \longrightarrow (V^*, \left\langle u - \alpha^* v , v - \alpha^* u \right\rangle)$ with a star-like polygon $\vartheta^*$ of $\vartheta^*V^*$ such that $T^{\vartheta^*}(v^*) = \vartheta^*V^*.$ Every star-like composite vector space $V^* \in P\omega_n^*$ can be uniquely decomposed as the sum of subspaces $w_i^* \leq W_{i+1}^*$ and $s_j^* \leq S_{j+1}^*$ such that $W_{i+1}^* + S_{j+1}^* \subseteq V^* \in P\omega_n^*.$ The study suggests that the research's findings be used to address issues in the  mathematical disciplines of genetics, engineering, code theory, and telecommunications.