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 \(A\).