  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 $A$.
