On adjoint invariant classes of shift operators

In the following text we compute the adjoint of weighted generalized shift operators over Hilbert spaces. We show for a conjugate invariant subset A of C, the additive semigroup generated by A∪\0\-weighted generalized shifts over Hilbert space H is adjoint invariant if and only if H is a finite dimensional Hilbert space or $0$ is not a limit point of A.