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.