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 $\mathbb{C}$, the additive semigroup generated by $A\cup \{0\}-$weighted generalized shifts over Hilbert space $\mathcal{H}$ is adjoint invariant if and only if $\mathcal{H}$ is a finite dimensional Hilbert space or $0$ is not a limit point of $A$.