Verify conditions of explicit formula solutions correlated with the operating system acting on a Hilbert space

Abstract

The requirements for obtaining a common solution of the system of Bounded operators’ equations, Ψ∗𝑋Ψ=Ω, Ψ𝑋∗Ψ∗=Ω, are presented in this article, where any operator on the Hilbert spaces 𝐻 is represented by 𝑋, and 𝐵 is a closed range operator. The inquiry results in the demonstration of a number of theorems that give specific descriptions for the solutions and requires have been necessary in the same time sufficient for their existence. As principal findings are the express solution for the operator equations in general additionally bounded answers for situations in which 𝐵 possesses a pseudo-inverse operator. For linear bounded operators, this response is given by the operator's pseudo-inverse on Hilbert spaces. Significant achievements include the generic formula solution of the equations has operators. Using pseudo inverse operators, we may achieve bounded solutions in situations where Ψ possesses a pseudo inverse operator, Further provide the subscriber solution for the next equations of operator Ψ∗𝑋Ψ=Ω1, Ψ𝑋Ψ∗=Ω2, Ψ𝑋∗Ψ∗=Ω3, Ψ∗𝑋∗Ψ=Ω4 where in (Ψ, Ω1, Ω2, Ω3, Ω4) are bounded noninvertible operators formed on Hilbert space. Likewise for varying the values of 𝑋, 𝑋0, we have given specific and comprehensive answers. See what makes the two solutions different. For each of the theorems, offer proofs.